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Results tagged with polynomials
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user 66323
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.
7
votes
2
answers
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Polynomials such that $|p(z)|\leq p(|z|)$
Let $p(x)=1+p_1x+p_2x^2+\cdots+p_nx^n$ be a polynomial with real coefficients and no positive zeros. Define
$$\mu(x)=\frac{xp'(x)}{p(x)}, \hspace{3ex} \sigma(x)=x \mu'(x).$$
Many years ago, as part of …
- 91.8k
10
votes
1
answer
520
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Conditions for a power of a polynomial to have no negative coefficients
On the other hand the condition $|p(z)|<p(|z|)$ for non-real and positive $z$ by itself is surely necessary (because it is satisfied by polynomials without negative coefficients, so if $p^n$ satisfies … As a related note, David Handelman has proved that in the (much more difficult) context of polynomials in several variables, if one power of $p$ has no negative coefficients, then $p^n$ will have no negative …
- 91.8k