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Results tagged with polynomials
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user 15506
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
1
vote
Every positive polynomial with rational coefficients is above a completely Q-factorized nonn...
The answer to your question is yes by the following lemma:
Let $f$ be a polynomial with rational coefficients which is (strictly) positive on the real line and has degree at least $4$. Then there is …
3
votes
Counting roots: multidimensional Sturm's theorem
The Hermite method for real root counting generalizes to the multivariate case if your system of polynomial inequalities has only a finite number of COMPLEX roots. See for example the article "Radical …
11
votes
Accepted
Relating a Polynomial equation to the characteristic equation of a Hermitian matrix
tridiagonal $A$ is essentially equivalent to Sturm's algorithm, see the recent interesting work of Ronan Quarez:
http://arxiv.org/pdf/0811.2365v1.pdf
If the coefficients of the monic polynomial are itself polynomials … Garding's notion of hyperbolic polynomials), then you can still use the Hermite matrix to do something, see the recent article of Netzer, Plaumann and Thom:
http://arxiv.org/abs/1108.4380 …
5
votes
Accepted
Application of polynomials with non-negative coefficients
You can also prove a lower bound on the exponent required, see:
Powers, Reznick: A new bound for Polya’s Theorem with applications to polynomials positive on polyhedra
This theorem can be used in representation …