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Results tagged with polynomials
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user 112641
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.
6
votes
Accepted
Bound on sum of coefficients of polynomials w.r.t a weighted integral
Consider the $(k,0)$ Jacobi polynomials $P_n$, which are orthogonal with respect to the weight $(1-x)^k$ on $[-1,1]$. …
- 1,593
3
votes
Accepted
Functions on rings and polynomials with coefficients in a certain kind of localisation
Yes (assuming $R$ is nonzero).
Suppose $a,b\in R\setminus\{0\}$ satisfy $ab=0$. Suppose further that $f=\frac{r_n}{s_n}x^n+\cdots+\frac{r_1}{s_1}x+\frac{r_0}{s_0}\in (S^{-1}R)[x]$ satisfies $f(0)=1$ …
- 1,593
9
votes
Accepted
Real-rootedness of some polynomials
In the notation of Fisk's "Polynomials, roots, and interlacing", $P_k(x+1)\underline\ll P_k(x)$. Let the roots of $P_{k+1}(x)$ be $b_1\le\cdots\le b_n$. …
- 1,593
3
votes
Accepted
Where can we find polynomial's root?
No, $R[x]/A$ does not always have a root.
Let $R=\mathbb{Z}\langle a,b\rangle$ be the free ring on two noncommuting elements $a$ and $b$. Let $\Psi=x^2+a\in R[x]$, and let $M$ and $A$ be the objects …
- 1,593
2
votes
From recursive polynomials to a $q$-series
Let $a_n(q)=b_n(q)-\binom{n+k}{k+1}$ and $a_\infty(q)=\lim_{n\to\infty} a_n(q)$. Then
$$\begin{align}a_n(q)&=\binom{n+k-1}{k}-\binom{n+k}{k+1}+(1+q^{n-1})\left(a_{n-1}(q)+\binom{n+k-1}{k+1}\right)\\&= …
- 1,593