Compute the discriminant sequence of a polynomial - Maple Help

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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

withRegularChains&colon;

withParametricSystemTools&colon;

R:=PolynomialRingx&comma;y&comma;t

R:=polynomial_ring

(1)

p:=x2&plus;tx&plus;y

p:=tx&plus;x2&plus;y

(2)

q:=yx2&plus;ty

q:=x2y&plus;ty

(3)

lp1:=DiscriminantSequencep&comma;x&comma;R

lp1:=1&comma;t24y

(4)

lp2:=DiscriminantSequencep&comma;q&comma;x&comma;R

lp2:=1&comma;y&comma;t2y22ty2&plus;2y3&comma;t5y3&plus;t4y36t3y4&plus;t2y54t2y4&plus;8ty54y6

(5)

See Also

BorderPolynomial, ComplexRootClassification , RealRootClassification, RegularChains

References

  

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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