DiscriminantSequence - Maple Help

RegularChains[ParametricSystemTools]

 DiscriminantSequence
 Compute the discriminant sequence of a polynomial

 Calling Sequence DiscriminantSequence(p, v, R) DiscriminantSequence(p, q, v, R)

Parameters

 R - polynomial ring p - polynomial of R q - polynomial of R v - variable of R

Description

 • When input is only one polynomial p, the result of this function call is the list of polynomials in R which is the discriminant sequence of p regarded as a univariate polynomial in v; otherwise the discriminant sequence of p and q.
 • For a univariate polynomial p of degree n, its discriminant sequence is a list of n polynomials in the coefficients of p. The signs of these polynomials determine the number of distinct complex (real) zeros of p. The discriminant sequence of two polynomials p and q, together with the discriminant sequence of p, can help determining the number of distinct real roots of p=0 such that q>0 or q<0. For the details, please see the reference listed below.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,t\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $p≔{x}^{2}+tx+y$
 ${p}{≔}{t}{}{x}{+}{{x}}^{{2}}{+}{y}$ (2)
 > $q≔y{x}^{2}+ty$
 ${q}{≔}{y}{}{{x}}^{{2}}{+}{t}{}{y}$ (3)
 > $\mathrm{lp1}≔\mathrm{DiscriminantSequence}\left(p,x,R\right)$
 ${\mathrm{lp1}}{≔}\left[{1}{,}{{t}}^{{2}}{-}{4}{}{y}\right]$ (4)
 > $\mathrm{lp2}≔\mathrm{DiscriminantSequence}\left(p,q,x,R\right)$
 ${\mathrm{lp2}}{≔}\left[{1}{,}{y}{,}{-}{{t}}^{{2}}{}{{y}}^{{2}}{-}{2}{}{t}{}{{y}}^{{2}}{+}{2}{}{{y}}^{{3}}{,}{{t}}^{{5}}{}{{y}}^{{3}}{+}{{t}}^{{4}}{}{{y}}^{{3}}{-}{6}{}{{t}}^{{3}}{}{{y}}^{{4}}{+}{{t}}^{{2}}{}{{y}}^{{5}}{-}{4}{}{{t}}^{{2}}{}{{y}}^{{4}}{+}{8}{}{t}{}{{y}}^{{5}}{-}{4}{}{{y}}^{{6}}\right]$ (5)

References

 Yang, L., "Recent advances in determining the number of real roots of parametric polynomials", J. Symb. Compt. vol. 28, pp. 225--242, 1999.