 HilbertPolynomial - Maple Help

Groebner

 HilbertSeries
 compute Hilbert series
 HilbertPolynomial
 compute Hilbert polynomial Calling Sequence HilbertSeries(J, X, s, characteristic=p) HilbertPolynomial(J, X, s, characteristic=p) Parameters

 J - a list or set of polynomials or a PolynomialIdeal X - (optional) a list or set of variables, a ShortMonomialOrder, or a MonomialOrder s - (optional) a variable to use for the series or polynomial p - (optional) characteristic Description

 • The HilbertSeries command computes the Hilbert series of the ideal generated by J, which is defined as ${\sum }_{n=0}^{\mathrm{\infty }}\left({c}_{n}-{c}_{n-1}\right){s}^{n}$ where ${c}_{n}$ is the dimension of the vector space of normal forms (with respect to J) of polynomials of degree less than or equal to $n$.  The output is a rational function of the form $H\left(s\right)=\frac{P\left(s\right)}{{\left(1-s\right)}^{d}}$ where $d$ is the HilbertDimension of J. The Hilbert polynomial is defined as the polynomial asymptotically equal to ${c}_{n}-{c}_{n-1}$.
 • In the case of skew polynomials, the invariants that are returned are those of the left ideal generated by J.
 • The variables of the system can be specified using an optional second argument X. If X is a ShortMonomialOrder then a Groebner basis of J with respect to X is computed. Be aware that if X is not a graded monomial order (that is, tdeg or grlex) then the result may be incorrect. By default, X is the set of all indeterminates not appearing inside a RootOf or radical when J is a list or set, or PolynomialIdeals[IdealInfo][Variables](J) if J is an ideal.
 • The variable for the Hilbert series or polynomial can be specified with an optional third argument s. If this is omitted, the global name 'Z' is used.
 • The optional argument characteristic=p specifies the ring characteristic when J is a list or set. This option has no effect when J is a PolynomialIdeal or when X is a MonomialOrder.
 • The algorithms for HilbertSeries and HilbertPolynomial use the leading monomials of a total degree Groebner basis for J. To access this functionality directly (as part of a program), make J the list or set of leading monomials. The commands will detect this case and execute their algorithms with minimal overhead.
 • Note that the hilbertseries and hilbertpoly commands are deprecated.  They may not be supported in a future Maple release. Examples

 > $\mathrm{with}\left(\mathrm{Groebner}\right):$
 > $F≔\left[{x}^{31}-{x}^{6}-x-y,{x}^{8}-z,{x}^{10}-t\right]$
 ${F}{≔}\left[{{x}}^{{31}}{-}{{x}}^{{6}}{-}{x}{-}{y}{,}{{x}}^{{8}}{-}{z}{,}{{x}}^{{10}}{-}{t}\right]$ (1)
 > $h≔\mathrm{HilbertSeries}\left(F,\left\{x,y,z,t\right\},s\right)$
 ${h}{≔}\frac{{{s}}^{{6}}{-}{2}{}{{s}}^{{5}}{-}{11}{}{{s}}^{{4}}{-}{9}{}{{s}}^{{3}}{-}{6}{}{{s}}^{{2}}{-}{3}{}{s}{-}{1}}{{-}{1}{+}{s}}$ (2)
 > $\mathrm{HilbertPolynomial}\left(F,\left\{x,y,z,t\right\},n\right)$
 ${31}$ (3)
 > $\mathrm{series}\left(h,s=0,10\right)$
 ${1}{+}{4}{}{s}{+}{10}{}{{s}}^{{2}}{+}{19}{}{{s}}^{{3}}{+}{30}{}{{s}}^{{4}}{+}{32}{}{{s}}^{{5}}{+}{31}{}{{s}}^{{6}}{+}{31}{}{{s}}^{{7}}{+}{31}{}{{s}}^{{8}}{+}{31}{}{{s}}^{{9}}{+}{O}{}\left({{s}}^{{10}}\right)$ (4)

The system below is not holonomic, in the sense that the Hilbert dimension is greater than the number of polynomial variables (x and y).

 > $\mathrm{with}\left(\mathrm{Ore_algebra}\right):$
 > $A≔\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\left[\mathrm{Dy},y\right],\mathrm{polynom}=\left\{x,y\right\}\right):$
 > $T≔\mathrm{MonomialOrder}\left(A,\mathrm{tdeg}\left(\mathrm{Dx},\mathrm{Dy},x,y\right)\right):$
 > $p≔{x}^{2}-x+y:$
 > $F≔\left[p\mathrm{Dx}+\frac{\partial }{\partial x}p,p\mathrm{Dy}+\frac{\partial }{\partial y}p\right]$
 ${F}{≔}\left[\left({{x}}^{{2}}{-}{x}{+}{y}\right){}{\mathrm{Dx}}{+}{2}{}{x}{-}{1}{,}\left({{x}}^{{2}}{-}{x}{+}{y}\right){}{\mathrm{Dy}}{+}{1}\right]$ (5)
 > $G≔\mathrm{Basis}\left(F,T\right)$
 ${G}{≔}\left[{\mathrm{Dy}}{}{{x}}^{{2}}{-}{\mathrm{Dy}}{}{x}{+}{\mathrm{Dy}}{}{y}{+}{1}{,}{\mathrm{Dx}}{}{{x}}^{{2}}{-}{\mathrm{Dx}}{}{x}{+}{\mathrm{Dx}}{}{y}{+}{2}{}{x}{-}{1}\right]$ (6)
 > $\mathrm{HilbertDimension}\left(F,T\right)$
 ${3}$ (7)
 > $p≔\mathrm{HilbertPolynomial}\left(F,T,n\right)$
 ${p}{≔}{{n}}^{{2}}{+}{3}{}{n}$ (8)
 > $h≔\mathrm{HilbertSeries}\left(F,T,s\right)$
 ${h}{≔}\frac{{{s}}^{{3}}{-}{{s}}^{{2}}{-}{s}{-}{1}}{{\left({-}{1}{+}{s}\right)}^{{3}}}$ (9)
 > $\mathrm{series}\left(h,s\right)$
 ${1}{+}{4}{}{s}{+}{10}{}{{s}}^{{2}}{+}{18}{}{{s}}^{{3}}{+}{28}{}{{s}}^{{4}}{+}{40}{}{{s}}^{{5}}{+}{O}{}\left({{s}}^{{6}}\right)$ (10)
 > $\mathrm{series}\left(\mathrm{add}\left(\genfrac{}{}{0}{}{p}{\phantom{n=i}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{p}}{n=i},i=0..6\right),s\right)$
 ${154}$ (11)

Neither x nor y can be eliminated.

 > $\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(F,\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left(\left[x\right],\left[\mathrm{Dx},\mathrm{Dy},y\right]\right)\right)\right),x\right)$
 $\left[\right]$ (12)
 > $\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(F,\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left(\left[y\right],\left[\mathrm{Dx},\mathrm{Dy},x\right]\right)\right)\right),y\right)$
 $\left[\right]$ (13)
 > $f≔\frac{1}{p}$
 ${f}{≔}\frac{{1}}{{{n}}^{{2}}{+}{3}{}{n}}$ (14)
 > $\left({x}^{2}-x+y\right)\left(\frac{{\partial }^{2}}{\partial y\partial x}f\right)+2\left(2x-1\right)\left(\frac{\partial }{\partial y}f\right)$
 ${0}$ (15)

The system becomes holonomic when we add the polynomial above. The Hilbert dimension is now 2 and both x and y can be eliminated.

 > $\mathrm{F2}≔\left[\mathrm{op}\left(F\right),\left({x}^{2}-x+y\right)\mathrm{Dx}\mathrm{Dy}+2\left(2x-1\right)\mathrm{Dy}\right]$
 ${\mathrm{F2}}{≔}\left[\left({{x}}^{{2}}{-}{x}{+}{y}\right){}{\mathrm{Dx}}{+}{2}{}{x}{-}{1}{,}\left({{x}}^{{2}}{-}{x}{+}{y}\right){}{\mathrm{Dy}}{+}{1}{,}\left({{x}}^{{2}}{-}{x}{+}{y}\right){}{\mathrm{Dx}}{}{\mathrm{Dy}}{+}{2}{}\left({2}{}{x}{-}{1}\right){}{\mathrm{Dy}}\right]$ (16)
 > $\mathrm{HilbertDimension}\left(\mathrm{F2},T\right)$
 ${2}$ (17)
 > $\mathrm{HilbertPolynomial}\left(\mathrm{F2},T,n\right)$
 ${4}{}{n}$ (18)
 > $h≔\mathrm{HilbertSeries}\left(\mathrm{F2},T,s\right)$
 ${h}{≔}\frac{{{s}}^{{2}}{+}{2}{}{s}{+}{1}}{{\left({-}{1}{+}{s}\right)}^{{2}}}$ (19)
 > $\mathrm{series}\left(h,s=0\right)$
 ${1}{+}{4}{}{s}{+}{8}{}{{s}}^{{2}}{+}{12}{}{{s}}^{{3}}{+}{16}{}{{s}}^{{4}}{+}{20}{}{{s}}^{{5}}{+}{O}{}\left({{s}}^{{6}}\right)$ (20)
 > $\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(\mathrm{F2},\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left(\left[x\right],\left[\mathrm{Dx},\mathrm{Dy},y\right]\right)\right)\right),x\right)$
 $\left[{4}{}{y}{}{{\mathrm{Dy}}}^{{2}}{+}{{\mathrm{Dx}}}^{{2}}{-}{{\mathrm{Dy}}}^{{2}}{+}{6}{}{\mathrm{Dy}}\right]$ (21)
 > $\mathrm{remove}\left(\mathrm{has},\mathrm{Basis}\left(\mathrm{F2},\mathrm{MonomialOrder}\left(A,\mathrm{lexdeg}\left(\left[y\right],\left[\mathrm{Dx},\mathrm{Dy},x\right]\right)\right)\right),y\right)$
 $\left[{2}{}{x}{}{\mathrm{Dy}}{-}{\mathrm{Dx}}{-}{\mathrm{Dy}}\right]$ (22)