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doc /// Node Key exponents (exponents, MonoidElement) (exponents, RingElement) Headline the exponents of a polynomial Usage exponents m Inputs m:{RingElement,MonoidElement} a monomial Outputs :List for a monomial, provides the list of exponents; for a polynomial, provides a list whose elements are the lists of exponents of the terms of @TT "f"@ Description Example R = ZZ/101[x,y,z]; exponents (f = x^2 - 7 + x*y*z^11 + y) leadMonomial f exponents leadMonomial f SourceCode (exponents, RingElement) SeeAlso content coefficients listForm standardForm monomials Node Key content (content, RingElement) (content, RingElement, RingElement) Headline the content of a polynomial Usage content f content(f, x) Inputs f:RingElement x:RingElement a variable in the ring of @TT "f"@ Outputs :Ideal the content of a matrix or polynomial Description Text The content is the ideal in the base ring generated by the coefficients. Example R = ZZ[x,y] content(4*x + 6*x^5) content(4*x + 6*x^5, x) content(4*x + 6*x^5, y) generator oo SourceCode (content, RingElement) (content, RingElement, RingElement) SeeAlso exponents coefficients listForm standardForm monomials Node Key listForm (listForm, RingElement) (listForm, MonoidElement) Headline convert to list form Usage listForm f Inputs f:{RingElement,MonoidElement} Outputs :List the polynomial or monomial in a form represented by nested lists. Description Text A monomial is represented by the list of its exponents. A polynomial is represented by lists of pairs @TT "(m, c)"@, one for each term, where @TT "m"@ is a list of exponents for monomial, and @TT "c"@ is the coefficient. Example R = ZZ/101[x,y,z]; listForm (x^2 - 7 + x*y*z^11 + y) SourceCode (listForm, RingElement) SeeAlso standardForm content coefficients exponents monomials Node Key standardForm (standardForm, RingElement) (standardForm, MonoidElement) Headline convert to standard form Usage standardForm f Inputs Outputs :HashTable the polynomial or monomial in a form involving hash tables. Description Text A polynomial is represented by hash tables in which the keys are hash tables representing the monomials and the values are the coefficients. The monomials themselves are represented by hash tables in which the keys are the variables and the values are the corresponding exponents. Example R = ZZ/101[x,y,z]; standardForm (x^2 - 7 + x*y*z^11 + y) SourceCode (standardForm, RingElement) SeeAlso listForm content coefficients exponents monomials /// document { Key => {coefficients, (coefficients,Matrix),(coefficients,RingElement), [coefficients,Monomials],[coefficients,Variables]}, Headline => "get the monomials and coefficients of a polynomial or matrix", Usage => "(M,C) = coefficients f", Inputs => { "f" => {"a one-row ", TO "Matrix", " with ", TT "n", " columns, say, or a ", TO "RingElement", ", to be interpreted as a one-by-one matrix. (A future implementation will handle matrices with more than one row.)" }, Variables => List => { "a list ", TT "v", " of variables. If a value for this option is not specified, all of the (top-level) variables are used." }, Monomials => {"a list or one-row matrix of monomials, each of which is formed using just variables in ", TT "v", "."} }, Outputs => { "M" => Matrix => { "either the value of the ", TT "Monomials", " option, if specified (converted to a one-row matrix, if necessary), or a one-row matrix of those monomials appearing in ", TT "f", " that involve just variables in ", TT "v", ", in descending order. Let ", TT "m", " denote the number of columns it has." }, "C" => Matrix => { "the ", TT "m", " by ", TT "n", " matrix ", TT "C", " such that ", TT "C_(i,j)", " is the coefficient in ", TT "f_(0,j)", " of the monomial ", TT "M_(0,i)", ". In other words, ", TT "C", " is the unique matrix not involving the (specified) variables such that ", TT "M*C == f", ", unless a value was specified for the ", TT "Monomials", " option that did not include all the monomials in the variables ", TT "v", " used by ", TT "f" } }, EXAMPLE lines /// R = QQ[a,b,c,d,e,f][x,y]; F = a*x^2+b*x*y+c*y^2 (M,C) = coefficients F ///, "The resulting matrices have the following property.", EXAMPLE lines /// M*C === matrix F ///, "The Sylvester matrix of two generic quadratic forms:", EXAMPLE lines /// G = d*x^2+e*x*y+f*y^2 P = matrix{{x*F,y*F,x*G,y*G}} (M,C) = coefficients P M*C === P ///, "We may give the monomials directly. This is useful if we are taking coefficients of several elements or matrices, and need a consistent choice of monomials.", EXAMPLE lines /// (M,C) = coefficients(P, Monomials=>{x^3,y^3,x^2*y,x*y^2}) ///, "If not all of the monomials are used, no error is signaled, but ", TT "M*C == P", " no longer holds.", EXAMPLE lines /// (M,C) = coefficients(P, Monomials=>{x^3,y^3}) M*C == P ///, SeeAlso => {monomials,content,exponents,listForm,standardForm} }