(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 758488, 16377]*) (*NotebookOutlinePosition[ 811644, 18080]*) (* CellTagsIndexPosition[ 810782, 18052]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ A Crash Course in Gr\[ODoubleDot]bner Bases \ \>", "Title"], Cell["\<\ Bruno Buchberger Special Semester on Gr\[ODoubleDot]bner Bases, RICAM / RISC, Linz, Austria February 6, 2006\ \>", "Subtitle"], Cell["", "Subsubtitle"], Cell[TextData[StyleBox["Copyright Bruno Buchberger 2006\n\nCopyright Note: \ This file may be copied, stored, and distributed subject to the following \ conditions:\n\n- The file is kept unchanged and complete including this \ copyright note.\n- A message is sent to bruno.buchberger@jku.at.\n- If the \ material is used, this talk should be cited appropriately.", FontColor->RGBColor[1, 0, 0]]], "Text"], Cell["Gr\[ODoubleDot]bner Bases: What and How?", "Subsubsubtitle"], Cell["Applications of Gr\[ODoubleDot]bner Bases", "Subsubsubtitle"], Cell["Discussion", "Subsubsubtitle"], Cell["Gr\[ODoubleDot]bner Bases: What and How?", "Subsubsubtitle"], Cell[TextData[StyleBox["Applications of Gr\[ODoubleDot]bner Bases", FontColor->GrayLevel[0.666667]]], "Subsubsubtitle"], Cell[TextData[StyleBox["Discussion", FontColor->GrayLevel[0.666667]]], "Subsubsubtitle"], Cell[CellGroupData[{ Cell["Motivation", "Section"], Cell[TextData[{ "Dozens of ", StyleBox["(difficult)", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["problems", FontColor->RGBColor[0, 0, 1]], " turned out to be ", StyleBox["reducible", FontColor->RGBColor[0, 0, 1]], " to the construction of Gr\[ODoubleDot]bner bases. (~ 600 papers, 10 \ textbooks, entry 13P10 in AMS index)." }], "Text", CellDingbat->"\[EmptyCircle]", CellMargins->{{50.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "This is based on the fact that Gr\[ODoubleDot]bner bases have many ", StyleBox["nice properties ", FontColor->RGBColor[0, 0, 1]], "(e.g. canonicality property, elimination property, syzygy problem)." }], "Text", CellDingbat->"\[EmptyCircle]", CellMargins->{{50.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "For the construction of Gr\[ODoubleDot]bner bases we have an ", StyleBox["algorithm", FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellDingbat->"\[EmptyCircle]", CellMargins->{{50.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "A \"beautiful\" theory: The notion of Gr\[ODoubleDot]bner bases and the \ algorithm is ", StyleBox["easy to explain, ", FontColor->RGBColor[0, 0, 1]], "but", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "correctness is based on a ", StyleBox["non-trivial", 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248}, ImageMargins->{{67.3125, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[StyleBox["This talk is based on the paper BB, \"Introduction \ to Gr\[ODoubleDot]bner Bases\", pp. 3-31, in this book.", FontColor->RGBColor[1, 0, 0]]], "Text"], Cell[" The presentation in the paper is more formal.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Linear Combination of Polynomials", "Section", CellTags->"division-reduction"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(f\_1\), "=", RowBox[{\(\(-2\) y\), "+", StyleBox[\(x\ y\), FontColor->RGBColor[0, 0, 1]]}]}], "\[IndentingNewLine]", RowBox[{\(f\_2\), "=", RowBox[{\(-x\^2\), "+", StyleBox[\(y\^2\), FontColor->RGBColor[0, 0, 1]]}]}]}], "Input"], Cell[BoxData[ \(\(-2\)\ y + x\ y\)], "Output"], Cell[BoxData[ \(\(-x\^2\) + y\^2\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Leading power products", FontColor->RGBColor[0, 0, 1]], ": w.r.t. an ordering of the power products (e.g. lexicographically, by \ total degreee or ...)" }], "Text"], Cell["\<\ (There are infinitely many \"admissible\" orderings for Gr\[ODoubleDot]bner \ bases theory that can be characterized by two easy axioms.)\ \>", "Text"], Cell[TextData[{ "Consider now the following linear combination of ", Cell[BoxData[ \(f\_1\)]], " and ", Cell[BoxData[ \(f\_2\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g = \((y)\)\ f\_1 + \ \(\((\(-x\) + 2)\) \(f\_2\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(y\ \((\(-2\)\ y + x\ y)\) + \((2 - x)\)\ \((\(-x\^2\) + y\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g = \((y)\)\ f\_1 + \ \((\(-x\) + 2)\) f\_2\ // Expand\)], "Input"], Cell[BoxData[ \(\(-2\)\ x\^2 + x\^3\)], "Output"] }, Open ]], Cell[TextData[{ "Observation: The leading power product ", Cell[BoxData[ \(x\^3\)]], " of g is \n neither a multiple of the leading power product x y \ of ", Cell[BoxData[ \(f\_1\)]], "\n nor a multiple of the leading power product ", Cell[BoxData[ \(y\^2\)]], " of ", Cell[BoxData[ \(f\_2\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Definition of Groebner Bases (BB 1965, Gordon 1899)", "Section"], Cell[TextData[{ "A set F of ", StyleBox["polynomials", FontColor->RGBColor[0, 0, 1]], " is called a ", StyleBox["Groebner basis", FontColor->RGBColor[0, 0, 1]], " (w.r.t. the chosen ordering of power products) iff the above phenomenon \ cannot happen, i.e. iff\n", " for all ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(f\_1\), "TraditionalForm"], ",", "...", ",", \(f\_m\)}], TraditionalForm]]], "\[Element] F and all polynomials ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(h\_1\), "TraditionalForm"], ",", "...", ",", \(h\_m\)}], TraditionalForm]]], ",\n the leading power product of ", Cell[BoxData[ \(TraditionalForm\`\(\(\(h\_1\ f\_1\)\(\ \)\(+\)\)\ ... \)\ + \ h\_m\ f\_m\)]], " \n is a multiple of the leading power product of \n \ at least one of the polynomials in F." }], "Text", CellFrame->True], Cell[TextData[{ "\nCounterexample: The Set ", Cell[BoxData[ \(TraditionalForm\`F = {f\_1, f\_2}\)]], " of the Above Example is ", StyleBox["not", FontColor->RGBColor[0, 0, 1]], " a Groebner basis." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Example of a Groebner Basis", "Section"], Cell[TextData[{ "The following set G (results from F by adding ", Cell[BoxData[ \(\(-2\)\ x\^2 + x\^3\)]], ") is a Groebner basis:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(G = {\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Input"], Cell[BoxData[ \({\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]], Cell["For example,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((1 + 3 y)\) \((\(-2\)\ x\^2 + x\^3)\) + \((8 x + 3 x\ y)\) \((\(-2\)\ y + x\ y)\) + \((2 - x - y\^2)\) \((\(-x\^2\) + y\^2)\) // Expand\)], "Input"], Cell[BoxData[ \(\(-4\)\ x\^2 + 2\ x\^3 - 16\ x\ y + 2\ x\^2\ y + 3\ x\^3\ y + 2\ y\^2 - 7\ x\ y\^2 + 4\ x\^2\ y\^2 - y\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((1)\) \((\(-2\)\ x\^2 + x\^3)\) + \((8 x)\) \((\(-2\)\ y + x\ y)\) + \((y)\) \((\(-x\^2\) + y\^2)\) // Expand\)], "Input"], Cell[BoxData[ \(\(-2\)\ x\^2 + x\^3 - 16\ x\ y + 7\ x\^2\ y + y\^3\)], "Output"] }, Open ]], Cell[TextData[{ "Why is it ", StyleBox["difficult to check", FontColor->RGBColor[0, 0, 1]], " whether a given F is a Groebner basis?" }], "Text"], Cell["How can we check whether a given F is a Groebner basis?", "Text"], Cell["\<\ How can we get an \"equivalent\" Groebner basis G for a given F (which may \ not be a Groebner basis)?\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The \"Main Theorem\" of Algorithmic Gr\[ODoubleDot]bner Bases Theory (BB \ 1965):\ \>", "Section"], Cell["\<\ \ \>", "Text"], Cell[TextData[{ StyleBox["F", FontSlant->"Italic"], " is a Gr\[ODoubleDot]bner basis \[DoubleLongLeftRightArrow] ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+\(f\_1, f\_2 \[Element] F\)\)]], " remainder[ ", StyleBox["F", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`S\[Dash]polynomial[f\_1, f\_2]\)]], "] = 0." }], "Text", CellFrame->True], Cell[TextData[{ StyleBox["Proof:", FontWeight->"Bold"], " Nontrivial. Combinatorial. Some details later.\n" }], "Text"], Cell[TextData[{ "The theorem ", StyleBox["reduces", FontColor->RGBColor[0, 0, 1]], " an ", StyleBox["infinite", FontColor->RGBColor[0, 0, 1]], " check to a ", StyleBox["finite", FontColor->RGBColor[0, 0, 1]], " check: Recall definition of \"F is a Gr\[ODoubleDot]bner basis\":" }], "Text"], Cell[TextData[{ " for all ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(f\_1\), "TraditionalForm"], ",", "...", ",", \(f\_m\)}], TraditionalForm]]], "\[Element] F and polynomials ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(h\_1\), "TraditionalForm"], ",", "...", ",", \(h\_m\)}], TraditionalForm]]], ",\n the leading power product of ", Cell[BoxData[ \(TraditionalForm\`\(\(\(h\_1\ f\_1\)\(\ \)\(+\)\)\ ... \)\ + \ h\_m\ f\_m\)]], " \n is a multiple of the leading power product of at least one of \ the polynomials in F." }], "Text"], Cell["\<\ The power of the Gr\[ODoubleDot]bner bases method is contained in this \ theorem and its proof.\ \>", "Text", CellFrame->True, CellTags->"power"] }, Open ]], Cell[CellGroupData[{ Cell["S-Polynomials", "Section"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(f\_1\), "=", RowBox[{\(\(-2\) y\), "+", StyleBox[\(x\ y\), FontColor->RGBColor[0, 0, 1]]}]}], "\[IndentingNewLine]", RowBox[{\(f\_2\), "=", RowBox[{\(-x\^2\), "+", StyleBox[\(y\^2\), FontColor->RGBColor[0, 0, 1]]}]}]}], "Input"], Cell[BoxData[ \(\(-2\)\ y + x\ y\)], "Output"], Cell[BoxData[ \(\(-x\^2\) + y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(S\[Dash]polynomial[f\_1, f\_2]\), "=", RowBox[{ RowBox[{ StyleBox["y", FontColor->RGBColor[0, 0, 1]], " ", \(f\_1\)}], "-", RowBox[{ StyleBox["x", FontColor->RGBColor[0, 0, 1]], " ", \(f\_2\), " "}]}]}]], "Input"], Cell[BoxData[ \(y\ \((\(-2\)\ y + x\ y)\) - x\ \((\(-x\^2\) + y\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(S\[Dash]polynomial[f\_1, f\_2]\), "=", RowBox[{ RowBox[{ RowBox[{ StyleBox["y", FontColor->RGBColor[0, 0, 1]], " ", \(f\_1\)}], "-", RowBox[{ StyleBox["x", FontColor->RGBColor[0, 0, 1]], " ", \(f\_2\)}]}], " ", "//", "Expand"}]}]], "Input"], Cell[BoxData[ \(x\^3 - 2\ y\^2\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The Algorithm 'remainder'", "Section"], Cell[TextData[{ "Roughly, remainder[ F, g] results from ", StyleBox["replacing power products in ", FontColor->RGBColor[0, 0, 1]], StyleBox["g", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" by a lower products ", FontColor->RGBColor[0, 0, 1]], "using the polynomials in ", StyleBox["F ", FontSlant->"Italic"], "until no more replacements are possible." }], "Text"], Cell[TextData[StyleBox["Example:", FontWeight->"Bold"]], "Text"], Cell["Consider, again,", "Text"], Cell[BoxData[{ \(\(f\_1 = \(-2\)\ x\^2 + x\^3;\)\), "\[IndentingNewLine]", \(\(f\_2 = \(-2\)\ y + x\ y;\)\), "\[IndentingNewLine]", \(\(f\_3 = \(-x\^2\) + y\^2;\)\)}], "Input"], Cell[BoxData[ \(\(F = {f\_1, f\_2, f\_3};\)\)], "Input"], Cell["and ", "Text"], Cell[BoxData[ \(\(g = x\ y - 3\ x\ y\^2;\)\)], "Input"], Cell["A \"reduction\" (\"division\") step on g w.r.t. F:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g\_1 = g\ + \((3\ x)\) f\_3\)], "Input", CellTags->"reduction-not-unique"], Cell[BoxData[ \(x\ y - 3\ x\ y\^2 + 3\ x\ \((\(-x\^2\) + y\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g\_1 = g\ + \((3\ x)\) f\_3 // Expand\)], "Input", CellTags->"reduction-not-unique"], Cell[BoxData[ \(\(-3\)\ x\^3 + x\ y\)], "Output", CellTags->"reduction-not-unique"] }, Open ]], Cell["A next division step w.r.t. F:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F = {f\_1, f\_2, f\_3}\)], "Input"], Cell[BoxData[ \({\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g\_2 = g\_1\ + \((\(-1\))\) f\_2\ // Expand\)], "Input"], Cell[BoxData[ \(\(-3\)\ x\^3 + 2\ y\)], "Output"] }, Open ]], Cell["A next division step w.r.t. F:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F = {f\_1, f\_2, f\_3}\)], "Input"], Cell[BoxData[ \({\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g\_3 = g\_2\ + \((3)\) f\_1\ // Expand\)], "Input"], Cell[BoxData[ \(\(-6\)\ x\^2 + 2\ y\)], "Output"] }, Open ]], Cell["\<\ This is the remainder of the division of g w.r.t. F because ...\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Remainder Algorithms are Available in all Math Systems", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialReduce[g, F, {y, x}]\)], "Input"], Cell[BoxData[ \({{0, 1 - 3\ y, \(-6\)}, \(-6\)\ x\^2 + 2\ y}\)], "Output"] }, Open ]], Cell[TextData[{ "Note: the remaindering algorithm can be extended to a \"", StyleBox["remaindering with co-factors", FontColor->RGBColor[0, 0, 1]], "\":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g\ - \ 0\ \ \ \ \((\(-2\)\ x\^2 + x\^3)\)\ \ - \ \((1 - 3\ y)\)\ \ \((\ \ \(-2\)\ y + x\ y)\)\ \ - \ \ \((\(-6\)\ )\) \((\(-x\^2\) + y\^2)\)\ // \ Expand\)], "Input"], Cell[BoxData[ \(\(-6\)\ x\^2 + 2\ y\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Now We Can ", StyleBox["Check", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " Groebnerianity" }], "Section"], Cell["Let's again look to the above example:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F = {f\_1, f\_2, f\_3}\)], "Input"], Cell[BoxData[ \({\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialReduce[f\_1\ y\ - \ \(f\_2\) x\^2, F, {y, x}]\)], "Input"], Cell[BoxData[ \({{0, 0, 0}, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialReduce[f\_1\ y\^2\ - \ \(f\_3\) x\^3, F, {y, x}]\)], "Input"], Cell[BoxData[ \({{4 + 2\ x + x\^2, \(-4\)\ y - 2\ x\ y, \(-8\)}, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialReduce[f\_2\ y\ - \ \(f\_3\) x, F, {y, x}]\)], "Input"], Cell[BoxData[ \({{1, 0, \(-2\)}, 0}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "The Problem of ", StyleBox["Constructing", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " Gr\[ODoubleDot]bner Bases" }], "Section"], Cell[TextData[{ "\nGiven ", StyleBox["F", FontSlant->"Italic"], ", find ", StyleBox["G", FontSlant->"Italic"], " s.t. Ideal(", StyleBox["F", FontSlant->"Italic"], ") = Ideal(", StyleBox["G", FontSlant->"Italic"], ") and ", StyleBox["G", FontSlant->"Italic"], " is a ", "Gr\[ODoubleDot]bner", " basis." }], "Text"], Cell[TextData[{ "(Ideal(F) := the set of all linear combinations ", Cell[BoxData[ \(TraditionalForm\`\(\(\(h\_1\ f\_1\)\(\ \)\(+\)\)\ ... \)\ + \ h\_m\ f\_m\)]], " \n \n with ", Cell[BoxData[ \(TraditionalForm\`f\_1\)]], ", ..., ", Cell[BoxData[ \(TraditionalForm\`f\_m\)]], " \[Element] F and ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(h\_1\), "TraditionalForm"], ",", "...", ",", \(h\_m\)}], TraditionalForm]]], " arbitrary polynomials.)" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "An Algorithm for ", StyleBox["Constructing", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " Gr\[ODoubleDot]bner Bases (BB 1965)" }], "Section"], Cell["Recall the main theorem:", "Text"], Cell[TextData[{ StyleBox["F", FontSlant->"Italic"], " is a Gr\[ODoubleDot]bner basis \[DoubleLongLeftRightArrow] ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+\(f\_1, f\_2 \[Element] F\)\)]], " remainder[ ", StyleBox["F", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`S\[Dash]polynomial[f\_1, f\_2]\)]], "] = 0." }], "Text", CellFrame->False, CellMargins->{{58.375, Inherited}, {Inherited, Inherited}}], Cell["\<\ Based on the main theorem, the problem can be solved by the following \ algorithm:\ \>", "Text", CellTags->"algorithm"], Cell[TextData[{ "Start with G:= F. \nFor any pair of polynomials ", Cell[BoxData[ \(TraditionalForm\`f\_1, f\_2 \[Element] G\)]], ":\n\n h := remainder[ ", StyleBox["G", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`S\[Dash]polynomial[f\_1, f\_2]\)]], "] \n \n If ", StyleBox["h ", FontSlant->"Italic"], "= 0, consider the next pair.\n \n If ", StyleBox["h", FontSlant->"Italic"], " \[NotEqual] 0, add ", StyleBox["h", FontSlant->"Italic"], " to ", StyleBox["G", FontSlant->"Italic"], " and iterate. " }], "Text", CellFrame->True], Cell[TextData[{ "The algorithm allows ", StyleBox["many refinements and variants", FontColor->RGBColor[0, 0, 1]], " which, however, are all based on the notion of ", StyleBox["S-polynomial ", FontColor->RGBColor[0, 0, 1]], "and variants of the main theorem." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Correctness and Termination of the Algorithm", "Section"], Cell[TextData[{ StyleBox["Correctness", FontColor->RGBColor[0, 0, 1]], ": Easy as soon as we know the main theorem." }], "Text"], Cell[TextData[{ StyleBox["Termination", FontColor->RGBColor[0, 0, 1]], ": by ", StyleBox["Dickson", FontColor->RGBColor[0, 0, 1]], "'s Lemma (Dickson 1913, BB 1970)." }], "Text"], Cell[TextData[{ " A sequence ", Cell[BoxData[ \(TraditionalForm\`p\_1, \ p\_2, \ ... \)]], " of power products with the property that, for all i < j, ", Cell[BoxData[ \(TraditionalForm\`p\_i\)]], " does not divide ", Cell[BoxData[ \(TraditionalForm\`p\_j\)]], ", must be finite." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Specializations", "Section"], Cell[TextData[{ "The ", StyleBox["Gr\[ODoubleDot]bner bases algorithm", FontColor->RGBColor[0, 0, 1]], ",\n\n for linear polynomials, specializes to ", StyleBox["Gauss' algorithm", FontColor->RGBColor[0, 0, 1]], ", and\n\n for univariate polynomials, specializes to ", StyleBox["Euclid's algorithm", FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellTags->"Gauss-algorithm"] }, Open ]], Cell[CellGroupData[{ Cell["Example", "Section"], Cell["Let's again look at", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(f\_1\), "=", RowBox[{\(\(-2\) y\), "+", StyleBox[\(x\ y\), FontColor->RGBColor[0, 0, 1]]}]}], "\[IndentingNewLine]", RowBox[{\(f\_2\), "=", RowBox[{\(-x\^2\), "+", StyleBox[\(y\^2\), FontColor->RGBColor[0, 0, 1]]}]}]}], "Input"], Cell[BoxData[ \(\(-2\)\ y + x\ y\)], "Output"], Cell[BoxData[ \(\(-x\^2\) + y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(F = {f\_1, f\_2}\)], "Input"], Cell[BoxData[ \({\(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]], Cell["F is not a Groebner basis.", "Text"], Cell[TextData[{ "The S-polynomial of ", Cell[BoxData[ \(f\_1, f\_2\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{\(S\[Dash]polynomial[f\_1, f\_2]\), "=", RowBox[{ RowBox[{ RowBox[{ StyleBox["y", FontColor->RGBColor[0, 0, 1]], " ", \(f\_1\)}], "-", RowBox[{ StyleBox["x", FontColor->RGBColor[0, 0, 1]], " ", \(f\_2\)}]}], " ", "//", "Expand"}]}]], "Input"], Cell[BoxData[ \(x\^3 - 2\ y\^2\)], "Output"] }, Open ]], Cell["Its remainder w.r.t. F is:", "Text"], Cell[BoxData[ \(\(-2\)\ x\^2 + \(\(x\^3\)\(.\)\)\)], "Input"], Cell["\<\ All the other S-polynomials have remainder 0. Hence, we arrived at a Groebner \ basis.\ \>", "Text"], Cell[TextData[{ "The Groebner basis algorithm is available now available in all math \ software systems, e.g. in ", StyleBox["Mathematica", FontSlant->"Italic"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(G = GroebnerBasis[F, {y, x}]\)], "Input"], Cell[BoxData[ \({\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Reduced Gr\[ODoubleDot]bner Bases", "Section"], Cell[TextData[{ "A set F of ", StyleBox["polynomials", FontColor->RGBColor[0, 0, 1]], " is called a ", StyleBox["reduced Gr\[ODoubleDot]bner basis", FontColor->RGBColor[0, 0, 1]], " (w.r.t. the chosen ordering of power products) iff \n F is a Gr\ \[ODoubleDot]bner bases and,\n for all f \[Element] F, \n \ remainder[ F-{f}, f] = f and\n f is monic." }], "Text", CellFrame->True], Cell["\<\ Algorithm for obtaining a reduced Gr\[ODoubleDot]bner basis: Compute a Gr\ \[ODoubleDot]bner basis and then \"auto-reduce\" the basis.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Extended Gr\[ODoubleDot]bner Basis Algorithm", "Section"], Cell["\<\ Keeps track of how the polynomials in the Gr\[ODoubleDot]bner basis G can be \ linearly combined from the polynomials in F.\ \>", "Text"], Cell[TextData[StyleBox["Gr\[ODoubleDot]bner Bases: What and How?", FontColor->GrayLevel[0.666667]]], "Subsubsubtitle"], Cell["Applications of Gr\[ODoubleDot]bner Bases", "Subsubsubtitle"], Cell[TextData[StyleBox["Discussion", FontColor->GrayLevel[0.666667]]], "Subsubsubtitle"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Applications are Based on Three Main Properties of Gr\[ODoubleDot]bner \ Bases\ \>", "Section"], Cell["Canonicality Property", "Text"], Cell["Elimination Property", "Text"], Cell["Syzygy Property", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Canonicality", "Section"], Cell[TextData[{ StyleBox["Remaindering", FontColor->RGBColor[0, 0, 1]], " modulo a Gr\[ODoubleDot]bner basis F is a \"", StyleBox["canonical simplifier", FontColor->RGBColor[0, 0, 1]], "\" for ", StyleBox["congruence", FontColor->RGBColor[0, 0, 1]], " modulo F:" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`f\( \[Congruent] \_F\)g\ \ \[DoubleLongLeftRightArrow]\ \ \ remainder[F, f] = remainder[F, g]\)]]], "Text", CellMargins->{{41.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "f ", Cell[BoxData[ \(TraditionalForm\`\(\( \[Congruent] \_\(\(F\)\(\ \ \)\)\)\(remainder[ F, f]\)\)\)]] }], "Text", CellMargins->{{41.3125, Inherited}, {Inherited, Inherited}}], Cell["\<\ \ \>", "Text"], Cell[TextData[{ "\"Second order\" canonicality: \"", StyleBox["Reduced Gr\[ODoubleDot]bner basis", FontColor->RGBColor[0, 0, 1]], "\" is a \"", StyleBox["canonical simplifier", FontColor->RGBColor[0, 0, 1]], "\" for \"", StyleBox["have same congruence", FontColor->RGBColor[0, 0, 1]], "\": " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`Ideal[ F]\ = \ \(\(Ideal[ G]\ \ \[DoubleLongLeftRightArrow]\ \ reduced\[Dash]Gr\ \[ODoubleDot]bner\[Dash]basis[F]\)\(=\)\)\)]], "reduced\[Dash]Gr\[ODoubleDot]bner\[Dash]basis[G]" }], "Text", CellMargins->{{41.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Ideal[F] ", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\ \)\)\)]], "Ideal[ reduced\[Dash]Gr\[ODoubleDot]bner\[Dash]basis[F] ]." }], "Text", CellMargins->{{41.3125, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Elimination Ideals", "Section"], Cell[TextData[{ "Let \[Precedes] be the lexicographic ordering defined by ", Cell[BoxData[ \(TraditionalForm\`\(\(x\_1\)\(\ \)\(\[Precedes]\)\(\ \)\(x\_2\)\(\ \ \ \)\)\)]], "\[Precedes] ... \[Precedes] ", Cell[BoxData[ \(TraditionalForm\`x\_n\)]], ". If F is a Gr\[ODoubleDot]bner basis w.r.t. \[Precedes], then" }], "Text"], Cell["\<\ If F is a Gr\[ODoubleDot]bner basis w.r.t. \[Precedes] \ then, for all i \[LessEqual] n,\ \>", "Text"], Cell[TextData[{ " Ideal[F] \[Intersection] ", Cell[BoxData[ FormBox[ RowBox[{"K", "[", RowBox[{ FormBox[\(x\_1\), "TraditionalForm"], ",", "\[Ellipsis]", ",", \(x\_i\)}], "]"}], TraditionalForm]]], " = Ideal[ F \[Intersection] ", Cell[BoxData[ FormBox[ RowBox[{"K", "[", RowBox[{ FormBox[\(x\_1\), "TraditionalForm"], ",", "\[Ellipsis]", ",", \(x\_i\)}], "]"}], TraditionalForm]]], " ]" }], "Text"], Cell[TextData[{ "\nThe \"", StyleBox["elimination ideals", FontColor->RGBColor[0, 0, 1]], "\" of an ideal can be easily computed if we have a Gr\[ODoubleDot]bner \ basis for the ideal." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Syzygy Property (Linear Syzygies)", "Section"], Cell[TextData[{ "Given a tuple \[LeftAngleBracket] ", Cell[BoxData[ \(TraditionalForm\`f\_1, \ \[Ellipsis], \ f\_m\)]], "\[RightAngleBracket] of polynomials. How can we obtain a finite basis for \ the set of all possible polynomial solutions (\"", StyleBox["syzygies", FontColor->RGBColor[0, 0, 1]], "\") \[LeftAngleBracket] ", Cell[BoxData[ \(TraditionalForm\`h\_1, \ \[Ellipsis], \ h\_m\)]], "\[RightAngleBracket] of the linear diophantine equation" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(h\_\(\(1\)\ \(\ \)\) . \ \ f\_\(\(1\)\(\ \)\) + \ \ \[Ellipsis]\ \ + \ h\_m\ . \ f\_m\ \ = \ \ \(\(0\)\(\ \ \ \ \)\(?\)\)\)\)\)], "Text"], Cell[TextData[{ "In the case that F:= { ", Cell[BoxData[ \(TraditionalForm\`f\_1, \ \[Ellipsis], \ f\_m\)]], "} is a Gr\[ODoubleDot]bner basis the following set of tuples is a finite \ basis for the infinite set of all syzgies:" }], "Text"], Cell[TextData[{ " consider all pairs ", Cell[BoxData[ \(TraditionalForm\`f\_i\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_j\)\(\ \)\)\)]], ": " }], "Text"], Cell[TextData[{ " m := LCM[ LPP[ ", Cell[BoxData[ \(TraditionalForm\`f\_i\)]], " ], LPP[ ", Cell[BoxData[ \(TraditionalForm\`f\_j\)]], "]], ", Cell[BoxData[ FormBox[ StyleBox[\(u\_i\), FontColor->RGBColor[1, 0, 0]], TraditionalForm]]], " := m / LPP[", Cell[BoxData[ \(TraditionalForm\`f\_i\)]], "], ", Cell[BoxData[ FormBox[ StyleBox[\(u\_j\), FontColor->RGBColor[1, 0, 0]], TraditionalForm]]], " := m / LPP[", Cell[BoxData[ \(TraditionalForm\`f\_j\)]], "]" }], "Text"], Cell[TextData[{ " \[LeftAngleBracket] ", Cell[BoxData[ \(TraditionalForm\`H\_1, \ \[Ellipsis], \ H\_m\)]], "\[RightAngleBracket] := the cofactors obtained by remaindering \n \ S-polynomial[ ", Cell[BoxData[ \(TraditionalForm\`f\_i\)]], ", ", Cell[BoxData[ \(TraditionalForm\`f\_j\)]], "] modulo F" }], "Text"], Cell[TextData[{ " \[LeftAngleBracket] ", Cell[BoxData[ \(TraditionalForm\`h\_1, \ \[Ellipsis], \ h\_i, \ \[Ellipsis], \ h\_j\)]], ", \[Ellipsis], ", Cell[BoxData[ \(TraditionalForm\`h\_m\)]], "\[RightAngleBracket] := \[LeftAngleBracket] -", Cell[BoxData[ FormBox[ RowBox[{\(H\_1\), ",", " ", "\[Ellipsis]", ",", " ", RowBox[{ StyleBox[\(u\_i\), FontColor->RGBColor[1, 0, 0]], " ", "-", " ", \(H\_i\)}], ",", " ", "\[Ellipsis]", ",", " ", RowBox[{ RowBox[{"-", " ", StyleBox[\(u\_j\), FontColor->RGBColor[1, 0, 0]]}], " ", "-", " ", \(H\_j\)}]}], TraditionalForm]]], ", \[Ellipsis], -", Cell[BoxData[ \(TraditionalForm\`H\_m\)]], "\[RightAngleBracket]" }], "Text"], Cell[TextData[{ "\n", StyleBox["Summarizing:", FontColor->RGBColor[0, 0, 1]], " the ", StyleBox["S-polynomials", FontColor->RGBColor[1, 0, 0]], " give a handle for obtaining a finite basis for the set of all ", StyleBox["syzygies", FontColor->RGBColor[0, 0, 1]], "!" }], "Text"], Cell[TextData[{ "(The ", StyleBox["inhomogeneous ", FontColor->RGBColor[0, 0, 1]], "equation" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(h\_\(\(1\ \)\(\ \)\) . \ \ f\_\(\(1\)\(\ \)\) + \ \ \[Ellipsis]\ \ + \ h\_m\ . \ f\_m\)\(\ \ \)\(=\)\(\ \ \)\(g\)\(\ \ \ \ \)\)\)\)], "Text"], Cell["\<\ can be solved by finding one solution of the inhomogeneous equation and \ adding the solutions of the homogeneous equations.)\ \>", "Text"], Cell[TextData[{ "(", StyleBox["In case { ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`f\_1, \ \[Ellipsis], \ f\_m\)], FontColor->RGBColor[0, 0, 1]], StyleBox["} is not a Gr\[ODoubleDot]bner basis", FontColor->RGBColor[0, 0, 1]], ", transform to Gr\[ODoubleDot]bner basis by the ", StyleBox["extended Gr\[ODoubleDot]bner basis algorithm", FontColor->RGBColor[0, 0, 1]], ", solve, and transform solutions back.)" }], "Text"], Cell[TextData[{ "(The case of ", StyleBox["several linear diophantine equations", FontColor->RGBColor[0, 0, 1]], " with polynomial coefficients can be reduced to the case of one equation. \ Alternatively, the entire Gr\[ODoubleDot]bner bases approach can be \ formulated for polynomial \"", StyleBox["modules", FontColor->RGBColor[0, 0, 1]], "\" instead of polynomial rings.)" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Application: Solving Polynomial Systems", "Section"], Cell[TextData[{ "Is based on the ", StyleBox["elimination property", FontColor->RGBColor[0, 0, 1]], " of Gr\[ODoubleDot]bner bases (w.r.t. lexicographic orderings)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["A Simple System of Equations", "Section"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(f\_1\), "=", RowBox[{\(\(-2\) y\), "+", StyleBox[\(x\ y\), FontColor->RGBColor[0, 0, 1]]}]}], "\[IndentingNewLine]", RowBox[{\(f\_2\), "=", RowBox[{\(-x\^2\), "+", StyleBox[\(y\^2\), FontColor->RGBColor[0, 0, 1]]}]}]}], "Input"], Cell[BoxData[ \(\(-2\)\ y + x\ y\)], "Output"], Cell[BoxData[ \(\(-x\^2\) + y\^2\)], "Output"] }, Open ]], Cell["Find x, y such that", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{\(\(-2\) y\), "+", StyleBox[\(x\ y\), FontColor->RGBColor[0, 0, 1]]}], "=", "0"}], "\[IndentingNewLine]", RowBox[{ RowBox[{\(-x\^2\), "+", StyleBox[\(y\^2\), FontColor->RGBColor[0, 0, 1]]}], "=", "0"}]}], "Input"], Cell["We compute", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(G = GroebnerBasis[F, {y, x}]\)], "Input"], Cell[BoxData[ \({\(-2\)\ x\^2 + x\^3, \(-2\)\ y + x\ y, \(-x\^2\) + y\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(-2\)\ x\^2 + x\^3 == 0, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0}, {x \[Rule] 0}, {x \[Rule] 2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \({G[\([2]\)], G[\([3]\)]} /. {x \[Rule] 2}\)], "Input"], Cell[BoxData[ \({0, \(-4\) + y\^2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\(-4\) + y\^2 == 0, y]\)], "Input"], Cell[BoxData[ \({{y \[Rule] \(-2\)}, {y \[Rule] 2}}\)], "Output"] }, Open ]], Cell[TextData[{ "All this is already implemented in the ", StyleBox["Mathematica", FontSlant->"Italic"], " general Solve function:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{f\_1 == 0, f\_2 == 0}, {x, y}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0, y \[Rule] 0}, {x 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Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 2]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 3]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 4]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 5]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 6]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 7]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 8]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 9]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 10]}, {z \[Rule] Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 11]}}\)], "Output"] }, Open ]], Cell["\<\ One can compute with these \"abstract roots\" like with \"square roots\". For \ example\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(G0 = G\ /. \ zsolexact1[\([2]\)] // Simplify\)], "Input"], Cell[BoxData[ \({Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1] + 4\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\ \^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^3 - 17\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^4 \ + 3\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + \ 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^5 - 45\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^6 \ + 60\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + \ 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^7 - 29\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^8 \ + 124\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 \ + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1]\^9 - 48\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^10 + 64\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^11 - 64\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^12, \(-22001\)\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1] + 14361\ y\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1] + 16681\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^2 + 26380\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^3 + 226657\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^4 + 11085\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^5 - 90346\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^6 - 472018\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^7 - 520424\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^8 - 139296\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^9 - 150784\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^10 + 490368\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^11, 43083\ y\^2 - 11821\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1] + 267025\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^2 - 583085\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^3 + 663460\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^4 - 2288350\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^5 + 2466820\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^6 - 3008257\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^7 + 4611948\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^8 - 2592304\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^9 + 2672704\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^10 - 1686848\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^11, 43083\ x - 118717\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1] + 69484\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ \ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^2 + 402334\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^3 + 409939\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^4 + 1202033\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^5 - 2475608\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^6 + 354746\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^7 - 6049080\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^8 + 2269472\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^9 - 3106688\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^10 + 3442816\ Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - \ 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, \ 1]\^11}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(G0 /. {Root[\(-1\) - 4\ #1\^2 + 17\ #1\^3 - 3\ #1\^4 + 45\ #1\^5 - 60\ #1\^6 + 29\ #1\^7 - 124\ #1\^8 + 48\ #1\^9 - 64\ #1\^10 + 64\ #1\^11 &, 1] \[Rule] \[Alpha]}\)], "Input"], Cell[BoxData[ \({\[Alpha] + 4\ \[Alpha]\^3 - 17\ \[Alpha]\^4 + 3\ \[Alpha]\^5 - 45\ \[Alpha]\^6 + 60\ \[Alpha]\^7 - 29\ \[Alpha]\^8 + 124\ \[Alpha]\^9 - 48\ \[Alpha]\^10 + 64\ \[Alpha]\^11 - 64\ \[Alpha]\^12, \(-22001\)\ \[Alpha] + 14361\ y\ \[Alpha] + 16681\ \[Alpha]\^2 + 26380\ \[Alpha]\^3 + 226657\ \[Alpha]\^4 + 11085\ \[Alpha]\^5 - 90346\ \[Alpha]\^6 - 472018\ \[Alpha]\^7 - 520424\ \[Alpha]\^8 - 139296\ \[Alpha]\^9 - 150784\ \[Alpha]\^10 + 490368\ \[Alpha]\^11, 43083\ y\^2 - 11821\ \[Alpha] + 267025\ \[Alpha]\^2 - 583085\ \[Alpha]\^3 + 663460\ \[Alpha]\^4 - 2288350\ \[Alpha]\^5 + 2466820\ \[Alpha]\^6 - 3008257\ \[Alpha]\^7 + 4611948\ \[Alpha]\^8 - 2592304\ \[Alpha]\^9 + 2672704\ \[Alpha]\^10 - 1686848\ \[Alpha]\^11, 43083\ x - 118717\ \[Alpha] + 69484\ \[Alpha]\^2 + 402334\ \[Alpha]\^3 + 409939\ \[Alpha]\^4 + 1202033\ \[Alpha]\^5 - 2475608\ \[Alpha]\^6 + 354746\ \[Alpha]\^7 - 6049080\ \[Alpha]\^8 + 2269472\ \[Alpha]\^9 - 3106688\ \[Alpha]\^10 + 3442816\ \[Alpha]\^11}\)], "Output"] }, Open ]], Cell["No more details in this talk!", "Text"], Cell["Alternatively, compute numerically:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(zsol = NSolve[G[\([1]\)] == 0, z]\)], "Input"], Cell[BoxData[ \({{z \[Rule] \(-0.3313043000789441`\) - 0.586934453864617`\ \[ImaginaryI]}, {z \[Rule] \ \(-0.3313043000789441`\) + 0.586934453864617`\ \[ImaginaryI]}, {z \[Rule] \ \(-0.29641346777662003`\) - 0.705328792560356`\ \[ImaginaryI]}, {z \[Rule] \ \(-0.29641346777662003`\) + 0.705328792560356`\ \[ImaginaryI]}, {z \[Rule] \ \(-0.16312357900403873`\) - 0.37694017948317526`\ \[ImaginaryI]}, {z \[Rule] \ \(-0.16312357900403873`\) + 0.37694017948317526`\ \[ImaginaryI]}, {z \[Rule] 0.`}, {z \[Rule] \(\(0.024891898039367064`\)\(\[InvisibleSpace]\)\) \ - 0.8917802016624393`\ \[ImaginaryI]}, {z \[Rule] \(\(0.024891898039367064`\)\ \(\[InvisibleSpace]\)\) + 0.8917802016624393`\ \[ImaginaryI]}, {z \[Rule] 0.46885190657576703`}, {z \[Rule] 0.670230792861074`}, {z \[Rule] 1.3928161982036307`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Gsubnum = G\ /. \ zsol[\([1]\)]\)], "Input"], Cell[BoxData[ \({1.3322676295501878`*^-15 + 9.71445146547012`*^-17\ \[ImaginaryI], \((\(-523.5194758552338`\) - 4967.646241304127`\ \[ImaginaryI])\) - \ \((\(\(4757.861053433717`\)\(\[InvisibleSpace]\)\) + 8428.965691949765`\ \[ImaginaryI])\)\ y, \ \((\(-7846.896476179643`\) - 8372.055369776856`\ \[ImaginaryI])\) + 43083\ y\^2, \((\(-16311.684476871265`\) + 16611.00043442221`\ \[ImaginaryI])\) + 43083\ x}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialGCD[Gsubnum[\([2]\)], Gsubnum[\([3]\)]]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ysol = NSolve[\ Gsubnum[\([2]\)] == 0, y]\)], "Input"], Cell[BoxData[ \({{y \[Rule] \(-0.4735346386353345`\) - 0.2051844321078939`\ \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ysol = NSolve[\ Gsubnum[\([3]\)] == 0, y]\)], "Input"], Cell[BoxData[ \({{y \[Rule] \(-0.4735346386353364`\) - 0.20518443210789353`\ \[ImaginaryI]}, {y \[Rule] \ \(\(0.4735346386353364`\)\(\[InvisibleSpace]\)\) + 0.20518443210789353`\ \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Theorem", FontWeight->"Bold"], " (Roider, Kalkbrener et al. 1990): It suffices to consider the poly in y \ with lowest degree." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(xsol = NSolve[\ Gsubnum[\([4]\)] == 0, x]\)], "Input"], Cell[BoxData[ \({{x \[Rule] \(\(0.37861069277606635`\)\(\[InvisibleSpace]\)\) - 0.3855581188501778`\ \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(F /. zsol[\([1]\)]\) /. ysol[\([1]\)]\) /. xsol[\([1]\)]\)], "Input"], Cell[BoxData[ \({\(-1.887379141862766`*^-15\) + 2.3592239273284576`*^-16\ \[ImaginaryI], 1.0269562977782698`*^-15 + 1.5265566588595902`*^-16\ \[ImaginaryI], \(-1.942890293094024`*^-15\) \ - 1.1102230246251565`*^-16\ \[ImaginaryI]}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Application: Invariant Theory", "Section"], Cell[TextData[{ StyleBox["A Question:", FontColor->RGBColor[0, 0, 1]], " Can " }], "Text", CellTags->"algebraic-relations"], Cell[CellGroupData[{ Cell[BoxData[ \(h = \(x\_1\^7\) x\_2 - x\_1\ x\_2\^7\)], "Input"], Cell[BoxData[ \(x\_1\%7\ x\_2 - x\_1\ x\_2\%7\)], "Output"] }, Open ]], Cell["be expressed as a polynomial in ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F = {x\_1\^2 + x\_2\^2, \(x\_1\^2\) x\_2\^2, \(x\_1\^3\) x\_2 - x\_1\ x\_2\^3}\)], "Input"], Cell[BoxData[ \({x\_1\%2 + x\_2\%2, x\_1\%2\ x\_2\%2, x\_1\%3\ x\_2 - x\_1\ x\_2\%3}\)], "Output"] }, Open ]], Cell["?", "Text"], Cell[TextData[{ "Note: These polynomials are fundamental invariants for ", Cell[BoxData[ \(TraditionalForm\`\[DoubleStruckCapitalZ]\_4\)]], ", i.e. a set of generators for the ring" }], "Text"], Cell[BoxData[ \(\(\({f \[Element] \[DoubleStruckCapitalC][x\_1, x\_2]\ \[VerticalSeparator] \ f \((x\_1, x\_2)\) = f \((\(-x\_2\), x\_1)\)}\)\(,\)\)\)], "Input"], Cell["i.e.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({x\_1\^2 + x\_2\^2, \(x\_1\^2\) x\_2\^2, \(x\_1\^3\) x\_2 - x\_1\ x\_2\^3} /. {x\_1 \[Rule] \(-x\_2\), x\_2 \[Rule] x\_1}\)], "Input"], Cell[BoxData[ \({x\_1\%2 + x\_2\%2, x\_1\%2\ x\_2\%2, x\_1\%3\ x\_2 - x\_1\ x\_2\%3}\)], "Output"] }, Open ]], Cell["\<\ and all invariants can be expressed as polynomials in these invariants.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Reduction to Groebner Bases Computation", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \({time, GB} = GroebnerBasis[{\(-i\_1\) + x\_1\^2 + x\_2\^2, \(-i\_2\) + \(x\_1\^2\) x\_2\^2, \(-i\_3\) + \(x\_1\^3\) x\_2 - x\_1\ x\_2\^3}, {x\_2, x\_1, i\_3, i\_2, i\_1}] // Timing\)], "Input"], Cell[BoxData[ \({0.`\ Second, {i\_1\%2\ i\_2 - 4\ i\_2\%2 - i\_3\%2, \(-i\_2\) + i\_1\ x\_1\%2 - x\_1\%4, i\_1\%2\ i\_3\ x\_1 - 2\ i\_2\ i\_3\ x\_1 - i\_1\ i\_3\ x\_1\%3 + i\_1\%2\ i\_2\ x\_2 - 4\ i\_2\%2\ x\_2, i\_1\%2\ x\_1 - 2\ i\_2\ x\_1 - i\_1\ x\_1\%3 + i\_3\ x\_2, \(-i\_1\)\ i\_3 + 2\ i\_3\ x\_1\%2 - i\_1\%2\ x\_1\ x\_2 + 4\ i\_2\ x\_1\ x\_2, \(-i\_3\)\ x\_1 - 2\ i\_2\ x\_2 + i\_1\ x\_1\%2\ x\_2, \(-i\_3\) - i\_1\ x\_1\ x\_2 + 2\ x\_1\%3\ x\_2, \(-i\_1\) + x\_1\%2 + x\_2\%2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialReduce[\(x\_1\^7\) x\_2 - x\_1\ x\_2\^7, GB, {x\_2, x\_1, i\_3, i\_2, i\_1}, MonomialOrder \[Rule] Lexicographic]\)], "Input"], Cell[BoxData[ \({{0, \(-i\_3\) - 1\/2\ i\_1\ x\_1\ x\_2 - x\_1\%3\ x\_2, 0, \(3\ i\_1\ x\_2\)\/4 - 1\/2\ x\_1\%2\ x\_2 + x\_2\%3\/2, i\_1 - x\_1\%2\/2 + \(3\ x\_2\%2\)\/4, \(3\ i\_1\ x\_1\)\/2 + x\_1\ x\_2\%2, x\_2\%4\/2, \(-\(1\/4\)\)\ i\_1\%2\ x\_1\ x\_2 - 1\/2\ i\_1\ x\_1\ x\_2\%3 - x\_1\ x\_2\%5}, i\_1\%2\ i\_3 - i\_2\ i\_3}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Theorem", FontWeight->"Bold"], " (Sweedler, Sturmfels et al. 1988): ", StyleBox["h", FontSlant->"Italic"], " can be represented in terms of ", StyleBox["I ", FontSlant->"Italic"], "iff remainder of ", StyleBox["h", FontSlant->"Italic"], " w.r.t. \"Groebner basis of ", StyleBox["I", FontSlant->"Italic"], " with slack variables\" is a polynomial in the slack variables (which \ gives the representation)." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(i\_1\%2\ i\_3 - i\_2\ i\_3 /. {i\_1 \[Rule] x\_1\^2 + x\_2\^2, i\_2 \[Rule] \(x\_1\^2\) x\_2\^2, i\_3 \[Rule] \(x\_1\^3\) x\_2 - x\_1\ x\_2\^3}\ // Expand\)], "Input"], Cell[BoxData[ \(x\_1\%7\ x\_2 - x\_1\ x\_2\%7\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(R = PolynomialReduce[\(x\_1\^6\) x\_2 - x\_1\ x\_2\^6, GB, {x\_2, x\_1, i\_3, i\_2, i\_1}, MonomialOrder \[Rule] Lexicographic]\)], "Input"], Cell[BoxData[ \({{0, \(i\_1\ x\_1\)\/2 - i\_1\ x\_2 - x\_1\%2\ x\_2, 0, \(3\ i\_1\)\/4 - x\_1\%2\/2 + x\_2\%2\/2, \(-\(x\_1\/4\)\) + \(3\ x\_2\)\/4, \(3\ i\_1\)\/4 + x\_1\ x\_2, x\_2\%3\/2, \(-\(1\/4\)\)\ i\_1\%2\ x\_1 - 1\/2\ i\_1\ x\_1\ x\_2\%2 - x\_1\ x\_2\%4}, \(-i\_1\%3\)\ x\_1 + 2\ i\_1\ i\_2\ x\_1 + 1\/2\ i\_1\ i\_3\ x\_1 + i\_1\%2\ x\_1\%3 - i\_2\ x\_1\%3 + 1\/2\ i\_3\ x\_1\%3 + 1\/2\ i\_1\ i\_2\ x\_2}\)], "Output"] }, Open ]], Cell[TextData[{ Cell[BoxData[ \(\(x\_1\^6\) x\_2 - x\_1\ x\_2\^6\)]], " can not be expressed by the fundamental invariants in I. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(x\_1\^6\) x\_2 - x\_1\ x\_2\^6 /. {x\_1 \[Rule] \(-x\_2\), x\_2 \[Rule] x\_1}\)], "Input"], Cell[BoxData[ \(x\_1\%6\ x\_2 + x\_1\ x\_2\%6\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Application: Automated (Dis-) Proving in Geometry", "Section"], Cell[TextData[{ StyleBox["Reduction of the Problem to Gr\[ODoubleDot]bner bases \ computation", FontColor->RGBColor[0, 0, 1]], ": \n\n Geo Theorem \[LongRightArrow] ( by \ coordinatization )\n ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+\(x, y, \ ... \)\)]], "( poly1(x,y,...)=0 \[And] ... \[Implies] poly(x,y,...)=0 ) \ \[LongRightArrow]\n \n ", Cell[BoxData[ \(TraditionalForm\`\[Not] \[Exists] \+\(x, y, \ ... \)\)]], " ( poly1(x,y,...)=0 \[And] ... \[And] poly(x,y,...)\[NotEqual]0 ) \ \[LongRightArrow]\n \n ", Cell[BoxData[ FormBox[ RowBox[{"\[Not]", UnderscriptBox["\[Exists]", RowBox[{"x", ",", "y", ",", " ", "...", ",", StyleBox["a", FontColor->RGBColor[1, 0, 0]]}]]}], TraditionalForm]]], "( poly1(x,y,...)=0 \[And] ... \[And] ", StyleBox["a", FontColor->RGBColor[1, 0, 0]], " . poly(x,y,...) - 1 = 0 ) " }], "Text"], Cell[TextData[{ "The latter question ", StyleBox["can be decided by the Gr\[ODoubleDot]bner basis method!", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ "The method is implemented in the ", StyleBox["Theorema System:", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell["\<\ B. B., C. Dupre, T. Jebelean, F. Kriftner, K. Nakagawa, D. Vasaru, W. \ Windsteiger. The Theorema Project: A Progress Report. In: Symbolic \ Computation and Automated Reasoning (Proceedings of CALCULEMUS 2000, \ Symposium on the Integration of Symbolic Computation and Mechanized \ Reasoning, August 6-7, 2000, St. Andrews, Scotland), M. Kerber and M. \ Kohlhase (eds.), A.K. Peters, Natick, Massachusetts, ISBN 1-56881-145-4, pp. \ 98-113.\ \>", "Text", CellMargins->{{46.625, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Example: Pappus Theorem", "Section"], Cell["What does the theorem say geometrically?", "Text", CellDingbat->"\[FilledSmallCircle]"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .68242 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0135387 0.00373483 -0.119385 0.00373483 [ [.04248 .35494 -5.8125 0 ] [.04248 .35494 5.8125 11.1875 ] [.09851 .13085 -9.21875 0 ] [.09851 .13085 9.21875 11.1875 ] [.37862 .46698 -5.75 0 ] [.37862 .46698 5.75 11.1875 ] [.45331 .0935 -9.1875 0 ] [.45331 .0935 9.1875 11.1875 ] [.97619 .66617 -6.03125 0 ] [.97619 .66617 6.03125 11.1875 ] 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SpanMaxSize->Infinity], GridBox[{ {\(f\^2 = 3/4 a\^2\)}, {\(b\^2 = \((a/2 - b)\)\^2 + \((a - f)\)\^2\)} }, ColumnAlignments->{Left}], StyleBox["}", ShowContents->False]}], "\[Implies]", \((b = 2 \((a - f)\))\)}], ")"}]}]], "Input"], Cell["The Transformation to a Groebner Basis Construction Problem:", "Text", FontWeight->"Bold"], Cell["This is equivalent to", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(\(\[Not] \[Exists] \)\+\(a, f, b\)\), RowBox[{"(", RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ {\(a \[NotEqual] 0\)}, {\(f\^2 = 3/4 a\^2\)}, {\(b\^2 = \((a/2 - b)\)\^2 + \((a - f)\)\^2\)}, {\(b \[NotEqual] 2 \((a - f)\)\)} }, ColumnAlignments->{Left}], StyleBox["}", ShowContents->False]}], ")"}]}], ","}]], "Input"], Cell["which is equivalent to", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(\(\[Not] \[Exists] \)\+\(a, f, b, \[Xi], \[Eta]\)\), RowBox[{"(", RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ {\(a\ \[Eta] = 1\)}, {\(f\^2 = 3/4 a\^2\)}, {\(b\^2 = \((a/2 - b)\)\^2 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Sturmfels):", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell["\<\ What is the minimum number of coins (e.g. p Pennies, n Nickels, d Dimes, q \ Quarters) for composing a given value, e.g. 117?\ \>", "Text", CellMargins->{{57.3125, Inherited}, {Inherited, Inherited}}], Cell["\<\ Reduction to Gr\[ODoubleDot]bner Bases Problem (C. Traverso et al. 1986):\ \>", "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "Code the integer ", StyleBox["values", FontColor->RGBColor[0, 0, 1]], " p, n, d, q as ", StyleBox["exponents", FontColor->RGBColor[0, 0, 1]], " of power products!" }], "Text", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Code the ", StyleBox["goal function", FontColor->RGBColor[0, 0, 1]], " as the (generalized) ", StyleBox["degree", FontColor->RGBColor[0, 0, 1]], " of the power products!" }], "Text", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Code the ", StyleBox["exchange rules ", FontColor->RGBColor[0, 0, 1]], "of the coins (the relations between the quantities) as ", StyleBox["polynomials", FontColor->RGBColor[0, 0, 1]], " consisting of power products:" }], "Text", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(F = {P\^5 - N, P\^10 - D, P\^25 - Q}\)], "Input", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({\(-N\) + P\^5, \(-D\) + P\^10, P\^25 - Q}\)], "Output", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[TextData[{ StyleBox["Now compute the Gr\[ODoubleDot]bner basis", FontColor->RGBColor[0, 0, 1]], " of F (w.r.t. degree ordering):" }], "Text", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(G = GroebnerBasis[F, MonomialOrder \[Rule] DegreeLexicographic]\)], "Input",\ CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({\(-D\) + N\^2, \(-D\^3\) + N\ Q, D\^2\ N - Q, \(-N\) + P\^5}\)], "Output", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[TextData[{ "Now you can be sure that, starting with any admissible solution (e.g. \ (p=17, n=10, d=5, q=0), ", StyleBox[" by reduction modulo ", FontColor->RGBColor[0, 0, 1]], StyleBox["G", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[", you will end up with a minimal solution", FontColor->RGBColor[0, 0, 1]], ":" }], "Text", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(PolynomialReduce[\(P\^17\) \(N\^10\) D\^5, G, , MonomialOrder \[Rule] DegreeLexicographic]\)], "Input", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \({{D\^9\ P\^17 + D\^8\ N\^2\ P\^17 + D\^7\ N\^4\ P\^17 + D\^6\ N\^6\ P\^17 + D\^5\ N\^8\ P\^17 + D\^4\ P\^17\ Q\^2 + P\^7\ Q\^4, \(-D\^7\)\ P\^17 - D\^4\ N\ P\^17\ Q - D\^2\ P\^17\ Q\^2, P\^17\ Q\^3, D\ P\^2\ Q\^4 + N\ P\^7\ Q\^4 + P\^12\ Q\^4}, D\ N\ P\^2\ Q\^4}\)], "Output", CellMargins->{{58.625, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[TextData[{ StyleBox["Answer", FontColor->RGBColor[0, 0, 1]], ": take 4 quarters, 1 dime, 1 nickel, 2 pennies." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Application: Symbolic Solution of Boundary Value Problems (Differential \ Equations)\ \>", "Section"], Cell["\<\ See work by Markus Rosenkranz, PhD thesis 2003 and RISC / RICAM Project \ \"Scientific Computing\".\ \>", "Text"], Cell["Will be presented in Workshop D2.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Other Applications", "Section"], Cell["\<\ Algebraic Geometry Coding Theory Cryptography Invariant Theory Integer Optimization Statistics Symbolic Integration Symbolic Summation Systems Theory\ \>", "Text"], Cell[TextData[StyleBox["Gr\[ODoubleDot]bner Bases: What and How?", FontColor->GrayLevel[0.666667]]], "Subsubsubtitle"], Cell[TextData[StyleBox["Applications of Gr\[ODoubleDot]bner Bases", FontColor->GrayLevel[0.666667]]], "Subsubsubtitle"], Cell["Discussion", "Subsubsubtitle"] }, Open ]], Cell[CellGroupData[{ Cell["How Difficult is the Construction of Gr\[ODoubleDot]bner Bases?", \ "Section"], Cell["Very Easy", "Text", CellMargins->{{25.3125, Inherited}, {Inherited, Inherited}}, FontColor->RGBColor[0, 0, 1], CellTags->"very-easy"], Cell["\<\ The structure of the algorithm is easy. The operations needed in the \ algorithm are elementary. \"Every high-school student can execute the \ algorithm.\" (See palm-top TI-98.) \ \>", "Text", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell["Very Difficult", "Text", CellMargins->{{25.3125, Inherited}, {Inherited, Inherited}}, FontColor->RGBColor[0, 0, 1], CellTags->"very-difficult"], Cell[TextData[{ "The inherent complexity of the problems that can be solved by the GB \ method (e.g. graph colorings) is \"exponential\". Hence, the worst-case \ complexity of the GB algorithm ", StyleBox["must", FontSlant->"Italic"], " be high. " }], "Text", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}, CellTags->"difficult"], Cell["Sometimes Easy", "Text", CellMargins->{{25.3125, Inherited}, {Inherited, Inherited}}, FontColor->RGBColor[0, 0, 1], CellTags->"sometimes-easy"], Cell["\<\ Mathematically interesting examples often have a lot of \"structure\" and, in \ concrete examples, GB computations can be reasonably, even surprisingly, \ fast. \ \>", "Text", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}, CellTags->"sometimes-easy"], Cell["Enormous Potential for Improvement", "Text", CellMargins->{{25.3125, Inherited}, {Inherited, Inherited}}, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "More ", StyleBox["mathematical ", FontSlant->"Italic"], "theorems can lead to drastic speed-up:" }], "Text", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}, CellTags->"improvements"], Cell["\<\ The use of \"criteria\" for eliminating the consideration of certain \ S-polynomials.\ \>", "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["p", FontSlant->"Italic"], "-adic approaches and floating point approaches." }], "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell["The \"Gr\[ODoubleDot]bner Walk\" approach.", "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell["The \"linear algebra\" approach. ", "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell["The \"numerics\" approach. ", "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell["Tuning of the algorithm:", "Text", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell["\<\ Heuristics, strategies for choosing orderings, selecting S-polynomials etc.\ \>", "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}, CellTags->"tuning"], Cell["Good implementation techniques.", "Text", CellDingbat->"\[GrayCircle]", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}], Cell[" A huge literature.", "Text", CellMargins->{{81.3125, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Why \"Gr\[ODoubleDot]bner\" Bases?", "Section"], Cell[TextData[{ "Professor ", StyleBox[" Wolfgang Gr\[ODoubleDot]bner", FontColor->RGBColor[1, 0, 0]], " (1899-1980) was my PhD thesis supervisor." }], "Text", CellTags->"Groebner-bases-name"], Cell[TextData[{ StyleBox["He gave me the problem", 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I introduced this name only in 1976, for honoring Gr\[ODoubleDot]bner, \ when people started to become interested in my work. " }], "Text"], Cell[TextData[StyleBox["My later contributions:", FontColor->RGBColor[0, 0, 1]]], "Text"], Cell["\<\ * the technique of criteria for eliminating unnecessary reductions * an abstract characterization of \"Gr\[ODoubleDot]bner bases rings\", see \ Workshop C \"Formal Gr\[ODoubleDot]bner Bases Theory\".\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["More Info on Gr\[ODoubleDot]bner Bases?", "Section"], Cell[TextData[StyleBox["Gr\[ODoubleDot]bner Bases 98 Conference:", FontColor->RGBColor[0, 0, 1]]], "Text", CellTags->"right-conference"], Cell[TextData[{ "B. B., F. Winkler. Gr\[ODoubleDot]bner", StyleBox[" Bases: Theory and Applications. 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_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn _[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn_[jn _[jn_[jn_[jn_[jn_[jn_[jn_[hE00003000004000160000500000P00017A4U30`0000`0000@0000 00000000000U0000300000P0080U0000300000D0080>000050000000000@00005000 \>"], "Graphics", ImageSize->{166.125, 248}, ImageMargins->{{67.3125, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}], Cell["This book contains tutorials and original papers.", "Text"], Cell["This book contains also:", "Text"], Cell[TextData[{ "B. B. ", StyleBox["Introduction to Gr\[ODoubleDot]bner Bases", FontSlant->"Italic"], ", pp. 3-31." }], "Text", CellDingbat->None, CellMargins->{{38.5, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "B. B. ", StyleBox["An Algorithmic Criterion for the Solvability of Systems of \ Algebraic Equations, pp. 540-560. ", FontSlant->"Italic"], "(English translation of the original paper from 1970, in which Gr\ \[ODoubleDot]bner bases were introduced.)" }], "Text", CellDingbat->None, CellMargins->{{40.5, Inherited}, {Inherited, Inherited}}], Cell["\<\ A continuation of this book is the special issue of the JSC on Gr\ \[ODoubleDot]bner bases edited by Q.N. Tran and F. Winkler, 2000.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Gr\[ODoubleDot]bner Bases on Your Desk and in Your Palm", "Section"], Cell[TextData[{ "GB implementations are contained in all the current math software systems \ like ", StyleBox["Mathematica ", FontSlant->"Italic"], "(see demo), Maple, Magma, Macsyma, Axiom, Derive, Reduce, Mupad, ..." }], "Text"], Cell[TextData[{ "Software systems specialized on Gr\[ODoubleDot]bner bases: ", StyleBox["RISA-ASIR (M. Noro, K. Yokoyama)", FontColor->RGBColor[0, 0, 1]], ", CoCoA, Macaulay, Singular, ..." }], "Text"], Cell[TextData[{ "Gr\[ODoubleDot]bner bases are now availabe on the ", StyleBox["TI-98 ", FontColor->RGBColor[0, 0, 1]], "(implemented in Derive)." }], "Text", CellTags->"TI-92"] }, Open ]], Cell[CellGroupData[{ Cell["Textbooks on Gr\[ODoubleDot]bner Bases", "Section"], Cell[TextData[{ "T. Kreuzer, L. Robbiano: ", StyleBox["Algorithmic Commutative Algebra I.", FontSlant->"Italic"], " Springer, Heidelber, 2000: Contains a list of all other, approx. 10, \ textbooks on GB." }], "Text"], Cell[TextData[{ "W.W.Adams, P. Loustenau. ", StyleBox["Introduction to ", FontSlant->"Italic"], "Gr\[ODoubleDot]bner", StyleBox[" Bases", FontSlant->"Italic"], ". Graduate Studies in Mathematics: Amer. Math. Soc., Providence, R.I., \ 1994." }], "Text"], Cell[TextData[{ "T.Becker, V.Weispfenning. ", "Gr\[ODoubleDot]bner", StyleBox[" Bases: A Computational Approach to Commutative Algebra.", FontSlant->"Italic"], " Springer, New York, 1993." }], "Text"], Cell[TextData[{ "D.Cox, J.Little, D.O'Shea. ", StyleBox["Ideals, Varieties, and Algorithms: An Introduction to \ Computational Algebraic Geometry and Commutative Algebra.", FontSlant->"Italic"], " Springer, New York, 1992." }], "Text"], Cell["....", "Text"], Cell["M. Maruyama. Gr\[ODoubleDot]bner Bases and Applications. 2002.", "Text", FontColor->RGBColor[0, 0, 1]], Cell["\<\ M. Noro, K. Yokoyama. Computational Fundamentals of Gr\[ODoubleDot]bner \ Bases. University of Tokyo Press, 2003.\ \>", "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["See papers data base and books in the secretary's \ office.", FontColor->RGBColor[1, 0, 0]]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Gr\[ODoubleDot]bner Bases on the Web", "Section"], Cell["\<\ Search. E.g. in the Research Index you obtain ~ 3000 citations.\ \>", "Text", CellTags->"Web"] }, Open ]], Cell[CellGroupData[{ Cell["Original Publications on Gr\[ODoubleDot]bner Bases", "Section"], Cell[TextData[StyleBox["See papers data base.", FontColor->RGBColor[1, 0, 0]]], "Text"], Cell["\<\ Approximately 600 papers appeared meanwhile on Gr\[ODoubleDot]bner bases.\ \>", "Text"], Cell["J of Symbolic Computation, in particular, special issues.", "Text"], Cell["ISSAC Conferences.", "Text"], Cell["Mega Conferences.", "Text"], Cell["ACA Conferences.", "Text"], Cell["...", "Text"], Cell["The essential additional original ideas in the literature:", "Text"], Cell[TextData[{ "Gr\[ODoubleDot]bner bases can be constructed w.r.t. arbitrary \"", StyleBox["admissible", FontColor->RGBColor[0, 0, 1]], "\" orderings (W. Trinks 1978)" }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Gr\[ODoubleDot]bner bases w.r.t. to \"lexical\" orderings have the ", StyleBox["elimination property ", FontColor->RGBColor[0, 0, 1]], "(W. Trinks 1978)" }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Gr\[ODoubleDot]bner bases can be used for computing ", StyleBox["syzygies", FontColor->RGBColor[0, 0, 1]], " and the S-polys generate the module of syzygies (G. Zacharias 1978)" }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "A given ", StyleBox["F", FontSlant->"Italic"], ", w.r.t. the ", StyleBox["infinitely", FontSlant->"Italic"], " many admissible orderings, has ", StyleBox["only ", FontColor->RGBColor[0, 0, 1]], StyleBox["finitely ", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["many Gr\[ODoubleDot]bner bases ", FontColor->RGBColor[0, 0, 1]], "and, hence, we can construct a \"universal\" Gr\[ODoubleDot]bner bases for \ ", StyleBox["F", FontSlant->"Italic"], " (L. Robbiano, V. Weispfenning, T. Schwarz 1988)" }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Starting from a Gr\[ODoubleDot]bner bases for ", StyleBox["F", FontSlant->"Italic"], " for ordering ", Cell[BoxData[ \(TraditionalForm\`O\_1\)]], " one can ", StyleBox["\"walk\", by changing the basis only slightly, to a basis for a \ \"nearby\"", FontColor->RGBColor[0, 0, 1]], " ordering ", Cell[BoxData[ \(TraditionalForm\`O\_2\)]], " and so on ... until one arrives at a Gr\[ODoubleDot]bner bases for a \ desired ordering ", Cell[BoxData[ \(TraditionalForm\`O\_k\)]], " (Kalkbrener, Mall 1995, Nam 2000)." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "Use ", StyleBox["arbitrary linear algebra algorithms ", FontColor->RGBColor[0, 0, 1]], "for the reduction (remaindering) process: (Faug\[EGrave]re 1997)." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "The ", StyleBox["numerics", FontColor->RGBColor[0, 0, 1]], " of Gr\[ODoubleDot]bner bases computation." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}], Cell[TextData[{ "... numerours ", StyleBox["applications", FontColor->RGBColor[0, 0, 1]], "," }], "Text", CellDingbat->"\[FilledSmallCircle]", CellMargins->{{75, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Research Topics", "Section"], Cell["\<\ the inner structure of Groebner bases: generalized Sylvester matrices\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["the numerics of GB computations", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["axiomatic characterization of Groebner rings", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["generalizations (e.g. non-commutative poly-rings)", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["speeding up the computation", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["\<\ Groebner bases for particular classes of ideals (avoid computation)\ \>", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["the study of admissible orderings", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["new applications", "Text", CellDingbat->"\[EmptySmallCircle]", CellMargins->{{68.625, Inherited}, {Inherited, Inherited}}], Cell["Appendix: Sketch of the Proof of the Main Theorem", "Subsubsubtitle"], Cell[TextData[{ StyleBox["Details see ", FontColor->RGBColor[0, 0, 1]], "BB, Introduction to Gr\[ODoubleDot]bner Bases, in BB, FW 1998, pp. 1 - \ 31." }], "Text"], Cell["Equivalent definition of Gr\[ODoubleDot]bner bases:", "Text"], Cell[TextData[{ " F is a Gr\[ODoubleDot]bner basis \[DoubleLongLeftRightArrow] ", Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \_F\)\)]], " has the Church-Rosser property." }], "Text", CellFrame->True], Cell[TextData[{ "\nf ", Cell[BoxData[ \(TraditionalForm\`\( \[Rule] \_F\)\)]], " g ... f reduces to g in one remaindering step using divisors from F." }], "Text"], Cell[TextData[{ " \[Rule] is Church-Rosser \[DoubleLongLeftRightArrow] ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+\(g\_1, g\_2\)\)]], " ( ", Cell[BoxData[ \(TraditionalForm\`g\_1\)]], " ", Cell[BoxData[ \(TraditionalForm\`\( \[LeftRightArrow] \^*\)\)]], Cell[BoxData[ \(TraditionalForm\`g\_2\)]], " \[DoubleLongRightArrow] ", Cell[BoxData[ \(TraditionalForm\`g\_1\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[DownArrow]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`g\_2\)]], " )" }], "Text"], Cell["\<\ Main Theorem:\ \>", "Text"], Cell[TextData[{ StyleBox["F", FontSlant->"Italic"], " is a Gr\[ODoubleDot]bner basis \[DoubleLongLeftRightArrow] ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+\(f\_1, f\_2 \[Element] F\)\)]], " remainder[ ", StyleBox["F", FontSlant->"Italic"], ", ", Cell[BoxData[ \(TraditionalForm\`S\[Dash]polynomial[f\_1, f\_2]\)]], "] = 0." }], "Text", CellFrame->True], Cell["\<\ Proof: \"\[DoubleLongRightArrow]\": Easy.\ \>", "Text"], Cell[TextData[{ "\nFor the direction \"\[DoubleLongLeftArrow]\" one can use the ", StyleBox["Newman Lemma ", FontColor->RGBColor[0, 0, 1]], "(Newman 1942). (For the version of the algorithm with criteria one needs \ the generalized Newman lemma by BB.) For Noetherian \[Rule]:" }], "Text"], Cell[TextData[{ " \[Rule] is Church-Rosser \[DoubleLongLeftRightArrow] ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+\(g\_1, g\_2, h\)\)]], " ( ", Cell[BoxData[ \(TraditionalForm\`g\_1\)]], " ", Cell[BoxData[ \(TraditionalForm\` \[LeftArrow] \)]], " h \[Rule] ", Cell[BoxData[ \(TraditionalForm\`g\_2\)]], " \[DoubleLongRightArrow] ", Cell[BoxData[ \(TraditionalForm\`g\_1\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\[DownArrow]\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`g\_2\)]], " )" }], "Text"], Cell["\<\ The proof of this lemma uses Noetherian induction. By using Newman's lemma in \ the proof of the main theorem, one takes induction out of the proof and is \ left with the specific technicalities of polynomial reduction.\ \>", "Text"], Cell[TextData[{ "\nHence, we have to consider, for arbitrary polynomials ", Cell[BoxData[ \(TraditionalForm\`g\_1, g\_2, h\)]], ", the situation that" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ FormBox[\(g\_1\), "TraditionalForm"], " ", FormBox[\( \[LeftArrow] \_F\), "TraditionalForm"], "h"}], \( \[Rule] \_F\), FormBox[\(g\_2\), "TraditionalForm"], " "}]}], TraditionalForm]]]], "Text"], Cell["\<\ and we have to show that we can always find a polynomial p such that\ \>", "Text"], Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\(g\_1\)\(\ \ \)\), "TraditionalForm"], \(\( \[Rule] \_F\)\^*\), " "}], TraditionalForm]]], "p ", Cell[BoxData[ \(TraditionalForm\`\(\( \[LeftArrow] \_F\)\^*\)\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(\(g\_2\)\(.\)\)\)]], " " }], "Text"], Cell[TextData[{ "By the assumption, there exist polynomials ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\)\(\ \)\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_2\)\(\ \ \)\)\)]], "in F such that h reduces w.r.t. ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\)\(\ \)\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`\(\(f\_2\)\(\ \)\)\)]], ". Let ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\(t\_1\)\(\ \)\(and\)\(\ \)\), "TraditionalForm"], \(t\_2\), " ", "be", " ", "the", " ", "power", " "}], TraditionalForm]]], "products in h on which these reductions work." }], "Text"], Cell[TextData[{ " h = ... + \[Square] ", Cell[BoxData[ \(TraditionalForm\`\(\(t\_1\)\(\ \ \ \ \)\)\)]], " + .... + \[Square] ", Cell[BoxData[ \(TraditionalForm\`\(\(t\_1\)\(\ \ \ \ \)\)\)]], "+ ....." }], "Text"], Cell[TextData[{ "\n - ", Cell[BoxData[ \(TraditionalForm\`\(\(u\_1\)\(\ \)\(f\_1\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\(u\_2\)\(\ \)\(f\_2\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)]] }], "Text"], Cell[TextData[{ " yields ", Cell[BoxData[ \(TraditionalForm\`g\_1\)]], " yields ", Cell[BoxData[ \(TraditionalForm\`g\_2\)]] }], "Text"], Cell[TextData[{ "\n", StyleBox["Cases ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(t\_1\)\(\ \)\(\[Precedes]\)\(\ \)\(t\_2\)\(\ \ \)\)\)], FontColor->RGBColor[0, 0, 1]], "and ", Cell[BoxData[ \(TraditionalForm\`\(\(t\_2\)\(\ \)\(\[Precedes]\)\(\ \)\(t\_1\)\(\ \ \)\)\)]], " easy (but not trivial!): by \"", StyleBox["semi-compatibility", FontColor->RGBColor[0, 0, 1]], "\" of polynomial reduction." }], "Text"], Cell[TextData[{ "\n", StyleBox["Cases t:= ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(t\_1\)\(\ \)\(=\)\(\ \)\(t\_2\)\(\ \)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox[":", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[TextData[{ " h = ... + \[Square] ", Cell[BoxData[ \(TraditionalForm\`\(\(t\)\(\ \ \ \ \)\)\)]], " + ....." }], "Text"], Cell[TextData[{ "\n - ", Cell[BoxData[ \(TraditionalForm\`\(\(u\_1\)\(\ \)\(f\_1\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\(\(u\_2\)\(\ \)\(f\_2\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)]] }], "Text"], Cell[TextData[{ "In this case t is a multiple of the LCM m of ", Cell[BoxData[ \(TraditionalForm\`LPP[f\_1]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`LPP[f\_2]\)]], ": t = v . m. " }], "Text"], Cell[TextData[{ "Since, by assumption of the theorem, the ", StyleBox["S-polynomial of ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`f\_1\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" and ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(f\_2\)\(\ \)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" can be reduced to 0", FontColor->RGBColor[0, 0, 1]], ", the reduction of m in the two essentially different ways (starting once \ by using ", Cell[BoxData[ \(TraditionalForm\`f\_1\)]], " and once by using ", Cell[BoxData[ \(TraditionalForm\`f\_2\)]], ") has a common successor." }], "Text"], Cell[TextData[{ "Hence, by \"", StyleBox["stability", FontColor->RGBColor[0, 0, 1]], "\" of polynomial reduction, by multiplication of all the steps by v, ", Cell[BoxData[ \(TraditionalForm\`g\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`g\_2\)]], "have a common successor." }], "Text"], Cell["My Recent Research Interest: Automated Theory Exploration", \ "Subsubsubtitle"], Cell[TextData[{ "For example: How can one ", StyleBox["invent", FontColor->RGBColor[1, 0, 0]], " (and verify) notions like \"S-polynomial\", theorems like the main \ theorem, and algorithms like the Gr\[ODoubleDot]bner bases algorithm ", StyleBox["automatically, ", FontColor->RGBColor[1, 0, 0]], "i.e. by algorithms that work on formulae." }], "Text"], Cell["For example, algorithm synthesis:", "Text"], Cell[" Given the specification P of a problem.", "Text"], Cell[TextData[{ " Find an algorithm A such that ", Cell[BoxData[ \(TraditionalForm\`\[ForAll] \+F\)]], " P[ F, A[F]]." }], "Text"], Cell["\<\ I succeeded to come up with a method which, for many P, yields A \ automatically. In particular, with this method, starting from the \ specification of the Gr\[ODoubleDot]bner bases construction problem: Given: F. Find: G such that G is finite G is a Gr\[ODoubleDot]bner basis Ideal[F] = Ideal[G], \ \>", "Text"], Cell["\<\ one arrives automatically at the notion of S-polynomials and the above Gr\ \[ODoubleDot]bner bases algorithm based on the notion of S-polynomials.\ \>", "Text"], Cell["For details see the recent publication", "Text"], Cell["\<\ B. Buchberger Towards the Automated Synthesis of a Gr\[ODoubleDot]bner Bases Algorithm RACSAM, 98/1 (Rev. Acad. Cienc., Spanish Royal Academy of Science), 98/1, pp. \ 65-75, 2005. \ \>", "Text", CellMargins->{{48, Inherited}, {Inherited, Inherited}}, FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "and the", StyleBox[" Workshop C \"Formal Gr\[ODoubleDot]bner Bases Theory\", ", FontColor->RGBColor[1, 0, 0]], "March 6-10, in the course of this special semester," }], "Text"] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1400}, {0, 967}}, ScreenStyleEnvironment->"Working", WindowToolbars->"RulerBar", WindowSize->{1392, 933}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.25, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. 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