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Results tagged with elimination-theory
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user 7410
Elimination theory is the study of necessary and sufficient conditions for polynomial equations (E) to have solutions.In the homogeneous case, if the number of variables is equal to the number of equations, this leads to the study of the Resultant (polynomial in the coefficients of (E), obtained by "eliminating" the variables ). In the general case, one get a Resultant ideal, generated by polynomial relations in the coefficients of the equations (E).
3
votes
Accepted
Discriminant of a composition of binary forms
If I did not mess up the adaptation to your special case, Theorem 3.31 for $n=2$ in "A computational approach to the discriminant of homogeneous polynomials" by Busé and Jouanolou
says that
$$
{\rm Di …
6
votes
Question on the integral $\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx$
The main theme here is the attempt to generalize the Isserlis-Wick Theorem for Gaussian integrals to the case of exponentials of polynomials of higher degree.
Let
$$
P(x)=a_n x^n+\cdots+a_1 x+a_0
$$
b …
13
votes
Accepted
Counting real zeros of a polynomial
By Remark 9.21 page 340 of the book by Basu, Pollack and Roy on real algebraic geometry the matrix $H$ is the expansion of the Bezoutiant of $P$ and $P'$ in the Horner basis of $P$ instead of the basi …
7
votes
Accepted
$n-1$ quadratic forms for $n$ variables
To reach a satisfactory understanding of the problem at hand, I think you need to learn about multidimensional resultants (see below for where to get started).
Working over the field $\mathbb{C}$, let …
3
votes
Accepted
Resultants and elimination theory
A bit long for a comment, so posted as an answer, although this is really a comment.
For 1) I would rather try the following. Assuming the $f$'s have the same degree in $x_1$,
introduce formal variabl …