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, ...

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1
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0answers
9 views

Is the restriction of a graded automorphism linearizable in characteristic zero?

This question follows up a previous one which was answered by Todd Leason. I want to impose two new requirements on the setup. Let $k$ be a characteristic zero field. Let $A=k[x_1,\dots,x_n]$ be the ...
7
votes
2answers
309 views

Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...
0
votes
0answers
26 views

Lowest upper bound of resultant [on hold]

Given two polynomials $f(x)=X^N+1$ and $g(x)= \sum_{i=0}^{N-1} s_i x^i$, where $s_i$'s are $\eta$-bit integers, let $res=resultant(f(x),g(x))$. What is the upper bound of $\log_2 res$?. Below my ...
4
votes
1answer
162 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
2
votes
1answer
55 views

Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
3
votes
1answer
80 views

Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points. Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...
3
votes
0answers
97 views

When a ring is a polynomial ring?

In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k ...
1
vote
0answers
57 views

Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$ Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
5
votes
0answers
67 views

Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group

Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...
5
votes
2answers
161 views

Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...
1
vote
1answer
88 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] ...
3
votes
0answers
114 views

Average nastiness of a Newton polytope

Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity: $$ P(x)= \left( \sum_{\alpha \in P} ...
3
votes
0answers
155 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots ...
5
votes
4answers
378 views

How do the roots of a polynomial change when another polynomial is added?

I need to obtain an analytical solution to an equation of the following form: $$ (x-a)(x-b)(x-c)=d(x-e)(x-f), $$ where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable. Of ...
3
votes
1answer
115 views

How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?

Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation) $$ ...
0
votes
0answers
33 views

Why $f_1,f_2,f_3$ don't have a common factor? [migrated]

Let $p(x,y)$ and $q(x,y)$ are polynomials in $x,y\in \mathbb{R}$. ($p,q\ne0$) $p$ and $q$ are coprime to each other ,($\frac{{\partial p}}{{\partial x}}=p_x$,$\frac{{\partial p}}{{\partial y}}=p_y$) ...
-2
votes
0answers
37 views

Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't have a common factor? [migrated]

Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} - q{(x,y)^2}$ and where $p(x, y)$ and $q(x, y)$ are real polynomials. Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial ...
1
vote
0answers
108 views

Values of Bernoulli polynomials at roots of unity

I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.
0
votes
0answers
31 views

Comparing product of positive affine functions over integers

Problem Let $f_i: \mathbb Z^n \mapsto \mathbb Z$ and $g_i: \mathbb Z^n \mapsto \mathbb Z$ affine functions and $\mathcal D \subseteq \mathbb Z^n$ a set on which they are all positive. Let $P$ and $Q$ ...
4
votes
0answers
97 views

Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y}.$$ I am ...
5
votes
2answers
153 views

$L^{\infty}$ polynomial approximation

In short: For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial? Jackson (1911) proved that there is a best approximating polynomial in the ...
-1
votes
0answers
102 views

One of the coefficients of $P(x)$ has absolute value $\geq |\alpha(\alpha-1)|$

Let $P \in \mathbb{Z}[x]$ be a polynomial with non-negative coefficients and degree $k$. Let $\alpha \leq -2$ be an integer such that: $P(\alpha) \equiv 0 (mod (\alpha^{k+1} -1))$ and ...
2
votes
0answers
58 views

Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write ...
3
votes
1answer
195 views

Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ... A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let ...
3
votes
0answers
95 views

An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
3
votes
1answer
63 views

Prime ideal ramified in extension if and only if certain polynomial divides another one?

Let $k$ be a field of characteristic $ \neq 2$, and let $f \in k[T]$ be a polynomial of degree $\ge 1$ which is square free. Let $K$ be the quadratic extension $k(T)(\sqrt{f})$ of $k(T)$. I know that ...
3
votes
1answer
127 views

Non-zero coefficients of primitive polynomials

Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$ be positive integers $\geq 2$. I want to prove that there exists a primitive polynomial $$F(x) = ...
3
votes
0answers
239 views

Why is solving polynomial systems NP hard?

Solving polynomial systems is known to be a NP hard problem; however it is not completely clear to me where this complexity comes from. My interest is in the case of systems of multivariate ...
1
vote
0answers
52 views

Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
6
votes
4answers
420 views

The coefficient of a specific monomial of the following polynomial

Let the real polynomial $$f_{a,b,c}(x_1,x_2,x_3)=(x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1},$$ where $a,b,c$ are nonnegative integers. Let $m_{a,b,c}$ be the coefficient of the monomial ...
3
votes
0answers
72 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
13
votes
0answers
392 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 ...
9
votes
0answers
181 views

Weyl Groups as Galois groups

I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of ...
0
votes
0answers
55 views

Extremum of the cyclic sum of polynomial ratios (same degree)

I've noticed a few times (probably nothing new) that cyclic sums (assuming $x, y, z > 0$) like: $\frac{x^2+y^2}{yz} + \frac{y^2+z^2}{zx} + \frac{z^2+x^2}{xy}$, where in each of the 3 ratios, all ...
0
votes
0answers
61 views

A representation problem on graphs

Given adjacency matrix $A\in\{0,1\}^{n\times n}$ of a graph denote $A(y,z)$ to be matrix where $0$ is replaced by $z$ and $1$ by $1-z$ on the non-diagonals (replace diagonals by $y$). Denote the ...
3
votes
1answer
187 views

system of complex equations

I am working on a system of complex equations The question is the following: Let $a_1,a_2,\ldots,a_N\in \mathbb{C}$ such that $$\sum_{j=1}^N \sum_{q=0}^{N-1-k} {N-1 \choose q} {N-1 \choose k+q} ...
0
votes
0answers
65 views

A question about divided differences

I want to ask a question about divided differences. Let $n\equiv0,1 \pmod 4$ is a positive integer. We know that for any polynomial $f\in \mathbb{Z}[x_1,x_2,\cdots,x_n]$, ...
2
votes
3answers
180 views

Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials. I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...
3
votes
0answers
65 views

Reducible polynomial compositions

Let $f(x)$ be a polynomial of degree $n\geq 2$ with integer coefficients. Then there always exists a polynomial $g(x)$ such that $f\circ g(x)$ is reducible. Namely, let $g(x)=f(x)+x$; then $f\circ ...
1
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0answers
67 views

Roots of generalized homogeneous polynomials

A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha ...
1
vote
1answer
327 views

Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...
1
vote
0answers
47 views

Solving the sextic equation using univariate analytic functions and arithmetic operations

Inspired by the top answer to this MO question, I would like to push the limit of the Hermite-Brioschi-Kronecker theorem. Suppose we only allow solutions to be expressed in terms of basic arithmetic ...
4
votes
0answers
248 views

Algebraic curves that enclose and exclude given points in the plane

Q1. Given two finite sets $R,G$ of points in $\mathbb{R}^2$, $|R|=r$ red points and $|G|=g$ green points, is it always possible to find a simple closed algebraic curve $C(x,y)=0$ that ...
1
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0answers
30 views

Particular functional equation for a polynomial

Let $P\in\mathbb C[X]$ be with degree $d\ge1$ and $q>1$. Can we find polynomials $P_i\in\mathbb C[X]$ and an integer $n\ge1$ such that $$\sum_{i=0}^nP_i(X^{p_i})P(q^iX)=0,$$ where $p_k$ denotes the ...
0
votes
1answer
54 views

Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials

Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand ...
5
votes
0answers
159 views

Solving a Laurent polynomial functional equation

I'm considering a set of functional equations: For a given $\phi(x)\in\mathbb {Z}[x,\frac {1}{x}] $ with $\phi(x)=\phi(\frac{1}{x}) $, $f(x)f(\frac {1}{x})+\phi (x)g(x)g(\frac {1}{x})=1, $ where ...
1
vote
1answer
60 views

Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as $diag(x)Ax=1$ $x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...
15
votes
1answer
359 views

Is the number of representations as the sum of two elements of a polynomial sequence always small?

Let $f(x) \in \mathbb{Z}[x]$ be a degree $d>1$ polynomial with integer coefficients. Define $$r(n) := | \{x,y \in \mathbb{Z} : f(x)+f(y) = n \}|. $$ My question is: Is it true that ...
4
votes
2answers
287 views

Inverse of a polynomial map

Ax-Grothendieck Theorem states that if $\mathbf K$ is an algebraically closed field, then any injective polynomial map $P:\mathbf K^n\longrightarrow \mathbf K^n$ is bijective. Question 1. What does ...
2
votes
1answer
65 views

Regularized integral and asymptotic expansion

Let $f:(0, \infty) \longrightarrow (0, \infty)$ be a monotonously increasing function (in fact, a step function) and let $P$ be a polynomial of degree $N$. Suppose I know that for some $k$, the limit ...