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.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\&colon;$

$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\&colon;$

$R≔\mathrm{PolynomialRing}\left(\left[x\&comma;y\&comma;t\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$p≔x^2+tx+y$

$p≔tx+x^2+y$

$q≔y{x}^2+ty$

$q≔y{x}^2+ty$

$\mathrm{lp1}≔\mathrm{DiscriminantSequence}\left(p\&comma;x\&comma;R\right)$

$\mathrm{lp1}≔\left[1\&comma;t^2-4y\right]$

$\mathrm{lp2}≔\mathrm{DiscriminantSequence}\left(p\&comma;q\&comma;x\&comma;R\right)$

$\mathrm{lp2}≔\left[1\&comma;y\&comma;-t^2{y}^2-2t{y}^2+2y^3\&comma;t^5{y}^3+t^4{y}^3-6t^3{y}^4+t^2{y}^5-4t^2{y}^4+8t{y}^5-4y^6\right]$

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