-- -*- coding: utf-8 -*- -- -*- coding: utf-8 -*- -- Copyright 1993-1999 by Daniel R. Grayson document { Key => (leadTerm, RingElement), Headline => "get the greatest term", Usage => "leadTerm f", Inputs => {"f" => "in a polynomial ring"}, Outputs => { RingElement => {"the lead term of ", TT "f", ""}}, "Each polynomial ring comes equipped with a ", TO2("monomial orderings", "monomial ordering"), " and this routine returns the lead (greatest) monomial and its coefficient. Recall that the default monomial order is the graded reverse lexicographic order.", -- Mike wanted this: TO "graded reverse lexicographic order" EXAMPLE { "R = QQ[a..d];", "leadTerm (3*b*c^2-d^3-1)", "S = QQ[a..d, MonomialOrder => Lex]", "leadTerm (3*b*c^2-d^3-1)" }, "Coefficients are included in the result:", EXAMPLE { "R = ZZ[a..d][x,y,z];", "leadTerm((a+b)*y^2 + (b+c)*x*z)" }, SeeAlso => {"leadCoefficient", "leadMonomial", "leadComponent"} } document { Key => {(leadTerm, Matrix),(leadTerm, GroebnerBasis),(leadTerm, Vector)}, Headline => "get the greatest term of each column", Usage => "leadTerm f", Inputs => {"f" => "in a polynomial ring"}, Outputs => { Matrix => {"the lead term matrix of ", TT "f", ""}}, "In Macaulay2, each free module over a polynomial ring comes equipped with a ", TO2("monomial orderings", "monomial order"), " and this routine returns the matrix whose ", TT "i", "-th column is the lead term of the ", TT "i", " th column of ", TT "f", ".", EXAMPLE lines /// R = QQ[a..d]; f = matrix{{0,a^2-b*c},{c,d}} leadTerm f ///, "Coefficients are included in the result:", EXAMPLE { "R = ZZ[a..d][x,y,z];", "f = matrix{{0,(a+b)*x^2},{c*x, (b+c)*y}}", "leadTerm f" }, "The argument ", TT "f", " can also be ", ofClass GroebnerBasis, ", in which case the lead term matrix of the generating matrix of ", TT "f", " is returned.", SeeAlso => {"leadCoefficient", "leadMonomial", "leadComponent"} } document { Key => (leadTerm, Ideal), Headline => "get the ideal of greatest terms", Usage => "leadTerm I", Inputs => {"I"}, Outputs => {{"The ideal of all possible lead terms of ", TT "I"}}, "Compute a ", TO2("Gröbner bases", "Gröbner basis"), " and return the ideal generated by the lead terms of the Gröbner basis elements.", EXAMPLE { "R = QQ[a..d];", "I = ideal(a*b-c*d, a*c-b*d)", "leadTerm I" }, EXAMPLE { "R = ZZ[a..d][x,y,z];", "I = ideal(a*x-b*y, x^3, y^3, z^3)", "leadTerm I" } } document { Key => (leadTerm, ZZ, RingElement), Headline => "get the lead polynomials using part of the monomial order", Usage => "leadTerm(n,f)", Inputs => {"n", "f" => "in a polynomial ring" }, Outputs => { RingElement => {"the lead term of ", TT "f", " using the first ", TT "n", " parts of the monomial order"}}, "Returns the sum of the terms of ", TT "f", " which are greatest using the first ", TT "n", " parts of the monomial order in the ring of ", TT "f", ".", -- Mike wanted this: " See ", TO "parts of monomial orders", " for an explanation.", PARA{}, "In the following example, the lead terms using the first part refers to all the monomials that have the lead monomial in the indeterminates ", TT "a", " and ", TT "b", ". This has a effect similar to selecting leadTerm in the ring ", TT "QQ[c,d][a,b]", ".", EXAMPLE lines /// R = QQ[a..d, MonomialOrder => ProductOrder{2,2}]; leadTerm(1, (c+d)*a^3 - c^100*a - 1) ///, SeeAlso => {selectInSubring} } document { Key => {(leadTerm, ZZ, Matrix), (leadTerm, ZZ, GroebnerBasis), (leadTerm, ZZ, Vector)}, Headline => "get the matrix of lead polynomials of each column", Usage => "leadTerm(n,f)", Inputs => {"n", "f" => "in a polynomial ring"}, Outputs => { Matrix => {"the lead term matrix of ", TT "f", " using the first ", TT "n", " parts of the monomial order"}}, "Returns the matrix whose ", TT "i", "-th column is the lead term of the ", TT "i", "-th column of ", TT "f", ", using the first ", TT "n", " parts of the monomial order. ", -- Mike wanted this: "See ", TO "parts of monomial orders", " for an explanation.", EXAMPLE { "R = QQ[x,y,z,a..d,MonomialOrder=>ProductOrder{3,4}];", "f = matrix{{0,x^2*(a+b)}, {a*x+2*b*y, y^2*(c+d)}}", "leadTerm(1,f)" }, SeeAlso => { "selectInSubring" } } document { Key => (leadTerm, ZZ, Ideal), Headline => "get the ideal of lead polynomials", Usage => "leadTerm(n,I)", Inputs => {"n", "I"}, Outputs => {{"The ideal of all possible lead polynomials of ", TT "I", " using the first ", TT "n", " parts of the monomial order"}}, "Compute a ", TO2("Gröbner bases", "Gröbner basis"), " and return the ideal generated by the lead terms of the Gröbner basis elements using the first n. ", -- Mike wanted this: "See ", TO "parts of monomial orders", " for an explanation.", EXAMPLE { "R = QQ[a..d,MonomialOrder=>ProductOrder{1,3}];", "I = ideal(a*b-c*d, a*c-b*d)", "leadTerm(1,I)" } } document { Key => leadTerm, Headline => "get the greatest term", "Every polynomial ring in Macaulay2 comes equipped with a monomial ordering. For ring elements and matrices, this function returns the greatest term in this order.", PARA{}, "For an ideal, a Gröbner basis is first computed, and the ideal of lead terms is returned.", PARA{}, "If an initial integer ", TT "n", " is specified, then the returned value contains the sum of all of the terms with the greatest value on the first ", TT "n", " ", "parts of the monomial order." -- Mike wanted this: TO2 ("parts of a monomial order", "parts of the monomial order"), "." } document { Key => {(borel, Matrix),borel,(borel, MonomialIdeal)}, Headline => "make a Borel fixed submodule", TT "borel m", " -- make a Borel fixed submodule", PARA{}, "Yields the matrix with the same target as the matrix ", TT "m", ", whose columns generate the smallest Borel fixed submodule containing the lead monomials of the columns of ", TT "m", ". If ", TT "m", " is a monomial ideal, then the minimal Borel fixed monomial ideal containing it is returned.", PARA{}, "For example, if R = ZZ/101[a..f], then", EXAMPLE { "R = ZZ/101[a..e]", "borel matrix {{a*d*e, b^2}}" } } document { Key => {singularLocus, (singularLocus, Ideal), (singularLocus, Ring)}, Headline => "singular locus", TT "singularLocus R", " -- produce the singular locus of a ring, which is assumed to be integral.", PARA{}, "This function can also be applied to an ideal, in which case the singular locus of the quotient ring is returned, or to a variety.", EXAMPLE lines /// singularLocus(QQ[x,y] / (x^2 - y^3)) singularLocus Spec( QQ[x,y,z] / (x^2 - y^3) ) singularLocus Proj( QQ[x,y,z] / (x^2*z - y^3) ) ///, PARA { "For rings over ", TO "ZZ", " the locus where the ring is not smooth over ", TO "ZZ", " is computed." }, EXAMPLE lines /// singularLocus(ZZ[x,y]/(x^2-x-y^3+y^2)) gens gb ideal oo /// } document { Key => {isSurjective,(isSurjective, Matrix)}, Headline => "whether a map is surjective", SeeAlso => "isInjective" } doc /// Key symmetricPower (symmetricPower,ZZ,Matrix) (symmetricPower,ZZ,Module) Headline symmetric power Usage symmetricPower(i,f) Inputs i:ZZ M:Matrix or @ofClass Module@ Outputs :Matrix or @ofClass Module@, the $i$-th symmetric power of the matrix or module $f$ Description Text There is currently one restriction: if $f$ is a matrix, then it must have only one row, and be a map of free modules, as in this example. Example R = ZZ/101[a..d] symmetricPower(2,vars R) Text If G --> F --> M --> 0 is a presentation for the module M = coker(f:G-->F), then symmetricPower(i,f) is the cokernel of the map symmetricPower(i-1,F) ** G --> symmetricPower(i,F). Example R = ZZ/101[a,b] symmetricPower(2,image vars R) SeeAlso exteriorPower basis /// document { Key => (exteriorPower,ZZ,Matrix), Headline => "exterior power of a matrix", Usage => "exteriorPower(i,f)\nexteriorPower_i f", Inputs => { "i", "f" }, Outputs => { { "the ", TT "i", "-th exterior power of ", TT "f", "."} }, EXAMPLE { "R = ZZ/2[x,y];", "f = random(R^3,R^{3:-1})", "exteriorPower_2 f" }, "The matrix may be a more general homomorphism of modules. For example,", EXAMPLE { "g = map(coker matrix {{x^2},{x*y},{y^2}}, R^3, id_(R^3))", "g2 = exteriorPower(2,g)", "target g2" }, SeeAlso => {(exteriorPower,ZZ,Module)} } document { Key => (exteriorPower,ZZ,Module), Headline => "exterior power of a module", Usage => "exteriorPower(i,M)\nexteriorPower_i M", Inputs => { "i", "M" }, Outputs => { {"the ", TT "i", "-th exterior power of ", TT "M", "."} }, EXAMPLE { "M = ZZ^5", "exteriorPower(3,M)" }, "When ", TT "i", " is ", TT "1", ", then the result is equal to ", TT "M", ". When ", TT "M", " is not a free module, then the generators used for the result will be wedges of the generators of ", TT "M", ". In other words, the modules ", TT "cover exteriorPower(i,M)", " and ", TT "exteriorPower(i,cover M)", " will be equal.", SeeAlso => {(exteriorPower,ZZ,Matrix)} } document { Key => exteriorPower, Headline => "exterior power", SeeAlso => {"minors", "det", "wedgeProduct"}, Subnodes => { TO [exteriorPower, Strategy] }, } document { Key => {(trace, Matrix),trace}, Headline => "trace of a matrix", TT "trace f", " -- returns the trace of the matrix f.", PARA{}, EXAMPLE { "R = ZZ/101[a..d]", "p = matrix {{a,b},{c,d}}", "trace p" }, } document { Key => {fittingIdeal,(fittingIdeal, ZZ, Matrix),(fittingIdeal, ZZ, Module)}, Headline => "Fitting ideal of a module", TT "fittingIdeal(i,M)", " -- the i-th Fitting ideal of the module M", PARA{}, EXAMPLE { "R = ZZ/101[x];", "k = coker vars R", "M = R^3 ++ k^5;", "fittingIdeal(3,M)", "fittingIdeal(8,M)" }, } document { Key => (symbol +, Module, Module), Headline => "sum of submodules", TT "M + N", " -- the sum of two submodules.", PARA{}, "The two modules should be submodules of the same module." } document { Key => ProjectiveHilbertPolynomial, Headline => "the class of all Hilbert polynomials", "For convenience, these polynomials are expressed in terms of the Hilbert polynomials of projective space.", PARA{}, "The functions ", TO "degree", " and ", TO "dim", " are designed so they correspond the degree and dimension of the algebraic variety that may have been used to produce the Hilbert polynomial.", EXAMPLE { "Z = Proj(QQ[x_0..x_12]/(x_0^3+x_12^3))", "hilbertPolynomial Z" }, Subnodes => { TO (symbol SPACE, ProjectiveHilbertPolynomial, ZZ), TO (degree, ProjectiveHilbertPolynomial), TO (dim, ProjectiveHilbertPolynomial), TO (euler, ProjectiveHilbertPolynomial), TO (hilbertSeries, ProjectiveHilbertPolynomial), }, } document { Key => (symbol SPACE, ProjectiveHilbertPolynomial, ZZ), Headline => "value of polynomial", TT "P i", " -- the value of a projective Hilbert polynomial ", TT "P", " at an integer ", TT "i", ".", PARA{}, EXAMPLE { "P = projectiveHilbertPolynomial 2", "apply(0 .. 12, i -> P i)", }, SeeAlso => ProjectiveHilbertPolynomial } document { Key => {projectiveHilbertPolynomial,(projectiveHilbertPolynomial, ZZ),(projectiveHilbertPolynomial, ZZ, ZZ)}, Headline => "Hilbert polynomial of projective space", TT "projectiveHilbertPolynomial n", " -- produces the projective Hilbert polynomial corresponding to projective space of dimension n.", BR{}, TT "projectiveHilbertPolynomial(n,d)", " -- produces the projective Hilbert polynomial corresponding to the graded ring of projective space of dimension n, but with its generator in degree -d.", PARA{}, SeeAlso => "ProjectiveHilbertPolynomial" } -- Local Variables: -- compile-command: "make -C $M2BUILDDIR/Macaulay2/m2 " -- End:

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