# 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.

1,713 questions
Filter by
Sorted by
Tagged with
26 views

### Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ be a natural number, let $p$ be an odd prime number with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. In the following, we work in the ring $\mathbb{F}_p[T]$...
182 views

21 views

### Name for product of square-free content of invariant factors of a matrix

Let $M(\lambda)$ be a possibly non-square polynomial matrix (over $\mathbb{R}$ or $\mathbb{C}$ is sufficient for me, but could be more general). By standard theory, it can be put into Smith normal ...
77 views

31 views

### cycloid-based polynomials

take the set of polynomials of the form $(a+b)^n$ and generalize them: let $P_f(a,b;n)$ be a sequence of polynomials where $f:(-c,c)\to \mathbb{R}^+$ is a function with $\int_{-c}^c f=1$ and $c$ can ...
92 views

256 views

### A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial

Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
66 views

### A density zero set of primes dividing the values of a non-constant integer polynomial

For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
36 views

### Bounds on degrees of minimal polynomials of infinite degree algebraic extension

If $E/F$ is algebraic extension of finite degree $n$, then if $\alpha \in E$ is an element, then the degree of minial polynomial $m_\alpha$ for $\alpha$ is at most $n$. Even better, $\deg m_\alpha$ ...
1k views

110 views

### Special elements of the Cremona group

After asking this MO question, I wish to ask about the following special case: Let $f$ be a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$. Is it possible to ...
238 views

### “Non-associative” standard polynomials

I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...
1k views

### How can I simplify this sum any further?

Recently I was playing around with some numbers and I stumbled across the following formal power series: $$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$ I was able ...
75 views

### Saturation with respect to $f$ of an ideal $I$

According to wikipedia, the saturation with respect to $f$ of an ideal $I$ in $R$ is the ideal $I:f^\infty:=\{g\in R:\exists k\in\mathbb{N}, f^kg\in I\}$. The important property of the saturation, ...
Consider the multivariate polynomial $$f(x_1,\ldots,x_m)=mk\sum_{i=1}^mx_i^2-mk(k-1)\sum_{i=1}^mx_i-\left(\sum_{i=1}^mx_i\right)^2,$$ for integers $m,k\ge2$. We are looking for integral zeros of $f$ ...
### Can we recover all $k$-minors of a square matrix from some of them?
This is a cross-post. Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of ...