# 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.

634 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
9k views

### Why polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
926 views

### Cubic function $\mathbb{Z}^2 \to \mathbb{Z}$ cannot be injective

It is easy to show, with an explicit construction, that a homogeneous cubic function $f: \mathbb{Z}^2 \to \mathbb{Z}$ is not injective. I am seeking a proof of the same result without the condition ...
581 views

### Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
639 views

### Three real polynomials

Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
791 views

### Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief:   When does a subset $S$ of the complex $n$th roots of unity have the property that $$\prod_{\alpha\, \in \,S} (z-\alpha)$$ gives a polynomial in $\mathbb R[z]$ with ...
505 views

### Zero curves of Tutte Polynomials?

There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a ...
598 views

211 views

### GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to four given spheres, an Apollonian problem in $\mathbb{R}^3$. The center of that fifth sphere is one of the $16$ possible solutions to ...
469 views

### Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
398 views

224 views

### Irreducibility of q-factorial plus 1

Let $q$ be a formal variable and for every positive integer $n$ let $$[n]_q! = 1 (1 + q)(1 + q + q^2) \dotsm (1 + q + \dotsb + q^{n-1})$$ be the $q$-factorial. Is it true that $[n]_q! + 1$ is an ...
617 views

### Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n+k$ unknowns ($n+k$ can be greater than 8), that is known to have a limited number of isolated solutions. I want to solve this system numerically, but if I plug it ...
229 views

### Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
196 views

### Galois group of polynomials related to Fibonacci and Catalan numbers

Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers. Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$. For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$. And another ...
442 views

### Descartes rule of signs for a noncommutative polynomial in matrix variables

Recently, while studying certain notions of "averaging" a set of input matrices, I obtained a nonlinear polynomial in matrix variables. A simple example is \begin{equation*} \mathcal{G}(X) := X^n - \...
359 views

241 views

283 views

### Is this a possible strengthening of the Lehmer conjecture?

Here is another possible refinement of the Lehmer conjecture. For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained ...
242 views

### Computing coefficients of polynomials from roots in $O(n\log{n})$ time
Suppose I have a univariate polynomial $p$ over a prime-order finite field $\mathbb{F}_q$ whose roots I know. Suppose that the roots of $p$ are always an $n$-sized subset of \$R=\{1,2,\dots,N\}, N <...