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Groebner

 Support
 compute the support of a polynomial

 Calling Sequence Support(f, T)

Parameters

 f - polynomial or a list or set of polynomials T - (optional) list or set of variables, MonomialOrder, or ShortMonomialOrder

Description

 • The Support command returns the list of the monomials present in a polynomial f. It automatically maps onto lists and sets, and can be used to quickly reveal the structure of a large Groebner basis.
 • An optional second argument T may be a list or set of variables or a MonomialOrder or ShortMonomialOrder.  If a monomial order is supplied then the resulting list of monomials is sorted in the monomial order.  If T is a list then the monomials are sorted in lexicographical order, or if T is a set then the monomials are not sorted.

Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $f≔87xy-56xy{z}^{2}-62{x}^{2}{z}^{3}+97x{y}^{3}z-73y{z}^{4}$
 ${f}{:=}{-}{62}{}{{x}}^{{2}}{}{{z}}^{{3}}{+}{97}{}{x}{}{{y}}^{{3}}{}{z}{-}{73}{}{y}{}{{z}}^{{4}}{-}{56}{}{x}{}{y}{}{{z}}^{{2}}{+}{87}{}{x}{}{y}$ (1)

Here, Support returns an unsorted list of monomials.

 > $\mathrm{Support}\left(f\right)$
 $\left[{{x}}^{{2}}{}{{z}}^{{3}}{,}{x}{}{{y}}^{{3}}{}{z}{,}{y}{}{{z}}^{{4}}{,}{x}{}{y}{}{{z}}^{{2}}{,}{x}{}{y}\right]$ (2)

The monomials are sorted in lexicographic order with $x>y>z$ .

 > $\mathrm{Support}\left(f,\left[x,y,z\right]\right)$
 $\left[{{x}}^{{2}}{}{{z}}^{{3}}{,}{x}{}{{y}}^{{3}}{}{z}{,}{x}{}{y}{}{{z}}^{{2}}{,}{x}{}{y}{,}{y}{}{{z}}^{{4}}\right]$ (3)

The monomials are sorted in graded lexicographic order with $x>y>z$ .

 > $\mathrm{Support}\left(f,\mathrm{grlex}\left(x,y,z\right)\right)$
 $\left[{{x}}^{{2}}{}{{z}}^{{3}}{,}{x}{}{{y}}^{{3}}{}{z}{,}{y}{}{{z}}^{{4}}{,}{x}{}{y}{}{{z}}^{{2}}{,}{x}{}{y}\right]$ (4)
 > $F≔\left[{x}^{2}+5{y}^{4},{x}^{8}-8{y}^{4},{z}^{8}+64{y}^{2}-{x}^{8}+100\right]$
 ${F}{:=}\left[{5}{}{{y}}^{{4}}{+}{{x}}^{{2}}{,}{{x}}^{{8}}{-}{8}{}{{y}}^{{4}}{,}{-}{{x}}^{{8}}{+}{{z}}^{{8}}{+}{64}{}{{y}}^{{2}}{+}{100}\right]$ (5)

Sometimes it is hard to see the structure of a Groebner basis. Support can quickly reveal the structure.

 > $G≔\mathrm{Groebner}[\mathrm{Basis}]\left(F,\mathrm{plex}\left(x,y,z\right)\right)$
 ${G}{:=}\left[{390625}{}{{z}}^{{64}}{+}{312500000}{}{{z}}^{{56}}{+}{109375000000}{}{{z}}^{{48}}{+}{21874994880000}{}{{z}}^{{40}}{+}{2734360643520000}{}{{z}}^{{32}}{+}{218740742876160000}{}{{z}}^{{24}}{+}{10935189454281097216}{}{{z}}^{{16}}{+}{312260738651419443200}{}{{z}}^{{8}}{+}{3897556646490972160000}{,}{264000003515625}{}{{z}}^{{56}}{+}{141792004560937500}{}{{z}}^{{48}}{+}{36975233639881250000}{}{{z}}^{{40}}{+}{5116635466117234680000}{}{{z}}^{{32}}{+}{535841466606972146400000}{}{{z}}^{{24}}{-}{4111661799133622084480000}{}{{z}}^{{16}}{+}{27769709934963826285883486208}{}{{z}}^{{8}}{+}{2305845823963169687985887117312}{}{{y}}^{{2}}{+}{3096625863672405027833148620800}{,}{-}{1320000017578125}{}{{z}}^{{56}}{-}{708960022804687500}{}{{z}}^{{48}}{-}{184876168199406250000}{}{{z}}^{{40}}{-}{25583177330586173400000}{}{{z}}^{{32}}{-}{2679207333034860732000000}{}{{z}}^{{24}}{+}{20558308995668110422400000}{}{{z}}^{{16}}{+}{41295655322303500444480000000}{}{{z}}^{{8}}{+}{288230727995396210998235889664}{}{{x}}^{{2}}{+}{2531291181350238048224000000000}\right]$ (6)
 > $\mathrm{Support}\left(G,\mathrm{plex}\left(x,y,z\right)\right)$
 $\left[\left[{{z}}^{{64}}{,}{{z}}^{{56}}{,}{{z}}^{{48}}{,}{{z}}^{{40}}{,}{{z}}^{{32}}{,}{{z}}^{{24}}{,}{{z}}^{{16}}{,}{{z}}^{{8}}{,}{1}\right]{,}\left[{{y}}^{{2}}{,}{{z}}^{{56}}{,}{{z}}^{{48}}{,}{{z}}^{{40}}{,}{{z}}^{{32}}{,}{{z}}^{{24}}{,}{{z}}^{{16}}{,}{{z}}^{{8}}{,}{1}\right]{,}\left[{{x}}^{{2}}{,}{{z}}^{{56}}{,}{{z}}^{{48}}{,}{{z}}^{{40}}{,}{{z}}^{{32}}{,}{{z}}^{{24}}{,}{{z}}^{{16}}{,}{{z}}^{{8}}{,}{1}\right]\right]$ (7)