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Results tagged with polynomials
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user 5371
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.
25
votes
Polynomials having a common root with their derivatives
The strongest result in this direction that I've heard of is Sudbery's theorem (which was
originally conjectured by Popoviciu and Erdös).
Theorem. Let $P(z)$ be a polynomial of degree $n\geq 2$ a …
- 22.3k
5
votes
Is there an intuitive explanation for an extremal property of Chebyshev polynomials?
By the way, $2^{1-n}T_n(x)$ minimizes all weighted $L^p$-norms
$$\left[\int_{-1}^{1}|P_n(x)|^p\frac{dx}{\sqrt{(1-x^2)}}\right]^{\frac{1}{p}},\quad 1\leq p\leq\infty,$$
over monic polynomials $P_n(x)$ … This book contains a survey of this and many other extremal properties of Chebyshev polynomials. …
- 22.3k