Compute the discriminant sequence of a polynomial
DiscriminantSequence(p, v, R)
DiscriminantSequence(p, q, v, R)
polynomial of R
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.
R ≔ PolynomialRing⁡x,y,t
R ≔ polynomial_ring
p ≔ x2+t⁢x+y
p ≔ t⁢x+x2+y
q ≔ y⁢x2+t⁢y
q ≔ x2⁢y+t⁢y
lp1 ≔ DiscriminantSequence⁡p,x,R
lp1 ≔ 1,t2−4⁢y
lp2 ≔ DiscriminantSequence⁡p,q,x,R
lp2 ≔ 1,y,−t2⁢y2−2⁢t⁢y2+2⁢y3,t5⁢y3+t4⁢y3−6⁢t3⁢y4+t2⁢y5−4⁢t2⁢y4+8⁢t⁢y5−4⁢y6
Yang, L., "Recent advances in determining the number of real roots of parametric polynomials", J. Symb. Compt. vol. 28, pp. 225--242, 1999.
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