# 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
93 views

280 views

### Linear homogenous polynomials that generates one quadratic polynomial

Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$. Assume that for every $i$ and ...
124 views

### Degrees of polynomials vanishing to various orders on a set of points

Suppose $X$ is a finite set of points in $\mathbb C^n$. Let $d_r$ denote the minimum degree of a polynomial vanishing to order $r$ at each point of $X$. By linear algebra, we know find can find a ...
222 views

### Reference request: integral formula for $\sum_{\text{roots }\lambda}e^{-|\lambda|^2}$

Consider a polynomial $f(z)=c\prod_m(z-\lambda_m)\in\mathbb{C}[z]$. I am mostly interested in the case where this actually lies in $\mathbb{R}[z]$, but that is not essential. I wanted to find a nice ...
204 views

### What does this permutation polynomial look like?

What is the number of terms of the unique multilinear polynomial $f\in\Bbb F_2[x_{1,1},\dots,x_{n,n}]$ in $n^2$ variables such that $f$ vanishes only on matrices that are permutations? Are there good ...
98 views

### Algebraic properties of harmonic polynomials

While studying a unique continuation property of some determinants, I encountered the following problem. Let $p_{ij}$ be real or complex harmonic polynomials (with homogeneous real and imaginary ...
96 views

### Reference request: maximal ratio of different norms of polynomials

Let us consider polynomials as functions on $[0,1]$, and so define \begin{align*} \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\ \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}. \end{align*} I am ...
176 views

575 views

### What does it mean polynomials share Newton polytope?

I have trouble understanding the connection between polynomials and Newton polytopes. I will try to make a short introduction to my problem and hope you will catch on. In the end I will ask questions. ...
445 views

### Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?

For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
146 views

### How to prove the following polynomial does not have root of a special form?

I'm working on a special kind of graphs. To prove some uniqueness, I need to prove that the polynomial $$x^{8}-7x^{6}+14x^{4}-8x^{2}+1$$ does not have any root of the form \...
76 views

### Can Davenport's estimate be extended to cubic polynomials with non-zero discriminant?

In 1961 Davenport showed that $H$ large enough there is a constant $c > 0$ such that $$\sum \lvert D(P) \rvert^{-1/2} < c H^2$$ where the sum is taken over the irreducible polynomials of ...
184 views

### Polynomials $P$ with integer roots near to $X^{\mathrm{deg}(P)}$

Let $d$ be a positive integer. My question is: can we then find a positive integer $r$ (dependent on $d$) with integers $\alpha_1, \alpha_2, ..., \alpha_r$ and $\beta_1, \beta_2, ..., \beta_r$ such ...
62 views

### Generically parameterize quartics in arbitrary dimension

Suppose $x=(x_1,\dots, x_d) \in \mathbb{R}^{d}$ for an arbitrary dimension $d$. Let $p(x_1,\dots, x_d)$ be a degree 4 polynomial and consider the quartic defined by $p(x_1, \dots, x_d)=0$. Is it ...
117 views

If we have a finite set of real-rooted polynomials of the same degree such that any two of them have a common interlacing then does it imply that this set has a common interlacer? Lemma $4.2$ (top of ...
139 views

123 views

52 views

### Is the difference between polyfits for two data series equivalent to the polyfit of the difference between the two data series?

Suppose that we have two series of data points $a(x)$ and $b(x)$ with the same domain of definition for $x$, and we fit two polynomial functions $f(x)$ and $h(x)$ (of the same order $n$) to them, ...
719 views

### Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive ...
130 views

### On a family of polynomials

Investigating experimentally a topic (somewhat related to Bernoulli convolutions), I came across families of polynomials and I wonder whether they belong to some well-known family. A closely related ...
182 views

### Proximity of solutions to system of degree two polynomials

Everything in this post is over the complex numbers. I would like to know if for every $\epsilon > 0$ there exists $\delta > 0$ such that the following holds for every $n$ and every $d$ which is ...
349 views

### Summation of double exponential series

Let $q \in (0,1)$ and consider the following summation: $$S(q,n) = \sum_{i=1}^n {q^2}^i$$ Is there a closed form expression or upper and lower bounds for $S(q,n)$? Specifically, I am looking for ...
187 views

### Minimal polynomial of a trigonometric number

I am trying to calculate the minimal polynomials of $h_{1}=-\cos(\pi/n)-\sqrt{\cos(2\pi/n)}$ and $h_{2}=-\cos(\pi/n)+\sqrt{\cos(2\pi/n)}$ when $n$ is odd. I think (and numerical calculations suggest ...
267 views

### Budan's theorem and real part of the roots

Budan's theorem gives an upper bound for the number of real roots of a real polynomial in a given interval $(a,b)$. This bound is not sharp (see the example in Wikipedia). My question is the ...
39 views

### Determining the Associated Sequence If Sheffer Conditions are not Met

A sheffer sequence $s_n(x)$ is formed by considering the generating function $$\sum_{k=0}^\infty s_k(x)\frac{t^k}{k!}=A(t)e^{xB(t)}$$ where $A$ is an invertible power series, and $B$ is a delta ...
247 views

### Polynomials on spaces of matrices

Let $\mathbb{P}^N$ be the projective space parametrizing $n\times n$ non-zero matrices modulo scalar multiplication, and let $\mathbb{P}^M\subset\mathbb{P}^N$ be the subspaces of symmetric matrices. ...
2k views

### What are the properties of this polynomial sequence?

Consider following polynomial sequence. $$\begin{cases}a_{-1}=0,~a_0=1, \\a_{n+1}=x \cdot a_n \pm a_{n-1}\end{cases}$$ Here $+$ or $-$ is taken in such a way that all coefficients in $a_n$ do not ...
317 views

### Stability of root-finding near the unit circle

It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (e....
348 views

### A congruence involving roots of unity

Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity. Computational evidence ...
138 views

### $g(a)$ divides monic $f(a)$ many times, is this possible?

Let $f,g \in \mathbb{Z}[x],\deg(f),\deg(g)>1$ and $f$ is monic. Assume $f$ and $g$ are coprime. For integer $a$ is it possible $g(a) \mid f(a)$ many times? Is it possible unbounded number of ...
1k views

### Do almost all systems of quadratic equations have solutions?

If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has ...
Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. By the primitive element theorem there exists $\alpha \in K$ such that $K = \mathbb{Q}(\alpha)$. Let \displaystyle S_K = \{\...