# Questions tagged [polynomials]

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.

1,534 questions
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### Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$ $$A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.$$ I'm interested in finding a ...
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### Bounded polynomial having coefficients that are bounded linearly in degree and number of variables

Let $P(\mathbf{x})$ be a bounded multivariate polynomial of degree at most $d$ (for my purposes it can be either coordinate degree or total degree) over $[-1,1]^n$, and assume $|P(\mathbf{x})| \leq 1$....
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### Prove that these are polynomials

Define the functions $$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k} \prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$ The numbers $\frac{2k+1}{n+k+1}\binom{2n}{n-k}$ ...
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### Dimension of $S$-units over $\mathbb{C}[x]$

Let $S=\{s_1,\ldots,s_n\}$ be a finite set of complex numbers. Consider the set of polynomials $$U=\{\,(x+s_1)^{k_1}\cdots(x+s_n)^{k_n}:\, 0\leq k_1+\cdots+k_n\leq H\}.$$ I am curious as to what is ...
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### Lacunary fully reducible polynomial over a finite field

The following problem is motivated by this MO question on rich directions determined by a set of a finite plane. Problem Does there exist a constant $C$ such that for all odd primes $p$ there is a ...
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### Real-rootedness of some polynomials

Denote the unsigned Stirling numbers of the first kind by $s(n,j)$. Question. Is it true that the polynomials $$P_n(x)=\sum_{j\geq0}s(n,j)\binom{x}j$$ have only real roots? Note. Obviously, ...
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### Structure of $k[X,Y,X^a/Y^b]$, name for such rings

Rings of the form $k[X,Y,\frac{X^a}{Y^b}]$, where $k$ is a finite field and $a$ and $b$ are relatively prime natural numbers, are showing up as residue rings in a problem I'm studying, and I'm ...
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### Nonzero subdeterminants conjecture: has anybody seen this anywhere?

I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is. Let $m\geq2$, $n\geq1$ be ...
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### Computing remainders modulo $\prod_{i\in S} (x-x_i)$ fast using FFT

Note: Originally asked on Math StackExchange here, without an answer. Figured I should try here, since this is a more research-level question. I am trying to implement a fast polynomial multipoint ...
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### Polynomial Eigenvalue Problem with few non-zero coefficients

Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...
Consider a proper map $F: \mathbb{C}^n\to \mathbb{R}^{2n-1}(n\geq 2)$ such that each component of $F$ is given by a sesquilinear form on $\mathbb{C}^n$. This to say, each component of $F$ can be ...