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MarkedPolynomialList

Description

A marked polynomial list is a list of polynomials in which the each polynomial has a distinguished term.

In gfan, the leading terms of polynomials are marked by writing them first. For example, the leading term of $y^2 + x^2 + z^2$ is $y^2$ as it appears first. In Macaulay 2, polynomials are sorted based on the term order of the ring and so distinguished terms are lost if they do not correspond to the current ring's term order.

i1 : QQ[x,y,z];
 
i2 : y^2 + x^2 + z^2

      2    2    2
o2 = x  + y  + z

o2 : QQ[x..z]

In gfanInterface, we represent marked Groebner bases using a list of leading terms and a second list of polynomials. Such a pair of lists is made into a MarkedPolynomialList by using the markedPolynomialList constructor.

i3 : QQ[x,y,z];
 
i4 : markedPolynomialList {{y^2, x^2}, {x^2 + y^2 + z^2, x^2 + y^2 + z^2}}

        2     2    2    2     2    2
o4 = {(y ) + x  + z , (x ) + y  + z }

o4 : MarkedPolynomialList

Many methods in gfanInterface require MarkedPolynomialLists as input and produce them as output.

For example, the method gfanMarkPolynomialSet takes a list of polynomials and a weight vector and returns a list of marked polynomials. In this case, the leading term is first computed using the weight vector and then lexicographic order to break ties.

i5 : QQ[x,y,z];
 
i6 : gfanMarkPolynomialSet({x*y^3+z^4, x^2*z^2 + y^3*z}, {-1,2,5})

        4       3    3      2 2
o6 = {(z ) + x*y , (y z) + x z }

o6 : MarkedPolynomialList

See also

Methods that use a marked polynomial list:

  • expression(MarkedPolynomialList) (missing documentation)
  • gfan(MarkedPolynomialList) -- see gfan -- all reduced Groebner bases of a polynomial ideal
  • gfanBuchberger(MarkedPolynomialList) -- see gfanBuchberger -- reduced Groebner basis with respect to some monomial order
  • gfanDoesIdealContain(MarkedPolynomialList,List) -- see gfanDoesIdealContain -- check ideal membership by the division algorithm
  • gfanGroebnerCone(Ideal,MarkedPolynomialList) -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
  • gfanGroebnerCone(List,MarkedPolynomialList) -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
  • gfanGroebnerCone(MarkedPolynomialList) -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
  • gfanGroebnerCone(MarkedPolynomialList,Ideal) -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
  • gfanGroebnerCone(MarkedPolynomialList,List) -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
  • gfanGroebnerCone(MarkedPolynomialList,MarkedPolynomialList) -- see gfanGroebnerCone -- polyhedral information about a Groebner cone
  • gfanHomogeneitySpace(MarkedPolynomialList) -- see gfanHomogeneitySpace -- homogeneity space of a list of polynomials
  • gfanHomogenize(MarkedPolynomialList,Symbol) -- see gfanHomogenize -- homogenize a list of polynomials with respect to a weight vector
  • gfanInitialForms(MarkedPolynomialList,List) -- see gfanInitialForms -- initial forms of polynomials with respect to a weight vector
  • gfanIsMarkedGroebnerBasis(MarkedPolynomialList) -- see gfanIsMarkedGroebnerBasis -- checks if a list of polynomials are a Groebner basis
  • gfanKrullDimension(MarkedPolynomialList) -- see gfanKrullDimension -- krull dimension
  • gfanLeadingTerms(MarkedPolynomialList) -- see gfanLeadingTerms -- leading terms of a list (or list of lists) of marked polynomials
  • gfanPolynomialSetUnion(List,MarkedPolynomialList) -- see gfanPolynomialSetUnion -- union of two lists of polynomials
  • gfanPolynomialSetUnion(MarkedPolynomialList,List) -- see gfanPolynomialSetUnion -- union of two lists of polynomials
  • gfanPolynomialSetUnion(MarkedPolynomialList,MarkedPolynomialList) -- see gfanPolynomialSetUnion -- union of two lists of polynomials
  • gfanSubstitute(MarkedPolynomialList,PolynomialRing) -- see gfanSubstitute -- rename the variables of a list of polynomials
  • gfanToLatex(MarkedPolynomialList) -- see gfanToLatex -- convert a list of polynomials to LaTeX
  • net(MarkedPolynomialList) (missing documentation)
  • RingMap MarkedPolynomialList (missing documentation)
  • texMath(MarkedPolynomialList) (missing documentation)
  • toString(MarkedPolynomialList) (missing documentation)

For the programmer

The object MarkedPolynomialList is a type, with ancestor classes List < VisibleList < BasicList < Thing.

The source of this document is in gfanInterface.m2:2722:0.