RegularChains[ParametricSystemTools]
DiscriminantSequence
Compute the discriminant sequence of a polynomial
Calling Sequence
Parameters
Description
Examples
References
DiscriminantSequence(p, v, R)
DiscriminantSequence(p, q, v, R)
R

polynomial ring
p
polynomial of R
q
v
variable of R
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.
$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$
$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\:$
$R\u2254\mathrm{PolynomialRing}\left(\left[x\,y\,t\right]\right)$
${R}{\u2254}{\mathrm{polynomial\_ring}}$
$p\u2254{x}^{2}+tx+y$
${p}{\u2254}{t}{}{x}{+}{{x}}^{{2}}{+}{y}$
$q\u2254y{x}^{2}+ty$
${q}{\u2254}{y}{}{{x}}^{{2}}{+}{t}{}{y}$
$\mathrm{lp1}\u2254\mathrm{DiscriminantSequence}\left(p\,x\,R\right)$
${\mathrm{lp1}}{\u2254}\left[{1}{\,}{{t}}^{{2}}{}{4}{}{y}\right]$
$\mathrm{lp2}\u2254\mathrm{DiscriminantSequence}\left(p\,q\,x\,R\right)$
${\mathrm{lp2}}{\u2254}\left[{1}{\,}{y}{\,}{}{{t}}^{{2}}{}{{y}}^{{2}}{}{2}{}{t}{}{{y}}^{{2}}{+}{2}{}{{y}}^{{3}}{\,}{{t}}^{{5}}{}{{y}}^{{3}}{+}{{t}}^{{4}}{}{{y}}^{{3}}{}{6}{}{{t}}^{{3}}{}{{y}}^{{4}}{+}{{t}}^{{2}}{}{{y}}^{{5}}{}{4}{}{{t}}^{{2}}{}{{y}}^{{4}}{+}{8}{}{t}{}{{y}}^{{5}}{}{4}{}{{y}}^{{6}}\right]$
Yang, L., "Recent advances in determining the number of real roots of parametric polynomials", J. Symb. Compt. vol. 28, pp. 225242, 1999.
See Also
BorderPolynomial
ComplexRootClassification
RealRootClassification
RegularChains
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