PolynomialIdeals/NumberOfSolutions - Maple Help

PolynomialIdeals

 NumberOfSolutions
 compute the number of solutions over the algebraic closure

 Calling Sequence NumberOfSolutions(J) NumberOfSolutions(G, tord)

Parameters

 J - a polynomial ideal G - list or set of polynomials; a Groebner basis tord - a monomial order

Description

 • The NumberOfSolutions command computes the number of solutions of a system over the algebraic closure of the coefficient field, including multiplicities. A zero-dimensional system has a finite number of solutions.
 • Let G be a Groebner basis for the ideal, then the number of solutions is equal to the number of monomials not divisible by a leading monomial of G.
 • Note that if the second calling sequence is used, NumberOfSolutions does not verify that G actually is a Groebner basis for the monomial order tord, and the result may be incorrect if it is not.
 • This function is part of the PolynomialIdeals package, and can be used in the form NumberOfSolutions(..) only after executing the command with(PolynomialIdeals).  However, it can always be accessed through the long form of the command using PolynomialIdeals[NumberOfSolutions](..).

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{2}-y,{y}^{3}+y+1⟩$
 ${J}{≔}⟨{{x}}^{{2}}{-}{y}{,}{{y}}^{{3}}{+}{y}{+}{1}⟩$ (1)
 > $\mathrm{IsZeroDimensional}\left(J\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{NumberOfSolutions}\left(J\right)$
 ${6}$ (3)

Observe that the generators of J are already a Groebner basis with respect to plex(x,y). The monomials not divisible by x^2 or y^3 are

 > $M≔\left[\mathrm{seq}\left(\mathrm{seq}\left({x}^{i}{y}^{j},j=0..2\right),i=0..1\right)\right]$
 ${M}{≔}\left[{1}{,}{y}{,}{{y}}^{{2}}{,}{x}{,}{x}{}{y}{,}{x}{}{{y}}^{{2}}\right]$ (4)

The same result can be obtained by using the generators directly.

 > $G≔\mathrm{Generators}\left(J\right)$
 ${G}{≔}\left\{{{x}}^{{2}}{-}{y}{,}{{y}}^{{3}}{+}{y}{+}{1}\right\}$ (5)
 > $\mathrm{NumberOfSolutions}\left(G,'\mathrm{plex}'\left(x,y\right)\right)$
 ${6}$ (6)
 > $K≔⟨{x}^{2}-y,yz-1⟩$
 ${K}{≔}⟨{{x}}^{{2}}{-}{y}{,}{y}{}{z}{-}{1}⟩$ (7)
 > $\mathrm{IsZeroDimensional}\left(K\right)$
 ${\mathrm{false}}$ (8)
 > $\mathrm{NumberOfSolutions}\left(K\right)$
 ${\mathrm{\infty }}$ (9)

References

 Cox, D.; Little, J.; and O'Shea, D. Using Algebraic Geometry. New York: Springer-Verlag, 1998.