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-...

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6 views

What are the fixed points of $\beta_j^{-n}[\alpha^n--\beta_j^{n-1}\mu_j-\beta_j^{n-2}\mu_j-…-\mu_j]$ for a fixed $j$

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= \beta_i x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. ...
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0answers
11 views

Constrained optimization with an integral [on hold]

I am trying to maximize the parameters $\alpha$ and $\beta$ of the following equation $ max_{\alpha, \beta} \sum_{n=1}^N ln (q(\lambda, \beta)) + \sum_{n=1}^N \sum_{k=1}^K \alpha_k \psi_k(\lambda)$ ...
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1answer
34 views

What are the fixed points of $\alpha^n-\mu_j$ for a fixed $j$?

Let us consider the polynomial ring $\Bbb C[x_1,...,x_s]$ and $\alpha(x_i)= x_i + \mu_i$ where $\mu_i \in \Bbb C$ are not all zero. Then $\alpha \in \mathrm{Aut}(\Bbb C[x_1,...,x_s])$. What are ...
0
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0answers
58 views

Normalizing factor in Reciprocity of traces of Frobenius with solutions of equations mod p

One kind of Reciprocity tells us that we can count solutions to polynomial equations over finite fields and relate them to traces of Galois Representations with some normalization factor. For ...
-1
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0answers
67 views

Finding the unique Nash equilibrium [closed]

$$ m^2(1-m)[(1+m)^2 - R] x^3 + [6m^2R + 12mR - 2m(1-m^2)(6+2m) - 4tm^2(1+m)^2] x^2 + [(1-m)(6+2m)^2 + 8tm(1+m)(6+2m)] x - 4t(6+2m)^2=0 $$ where: m ∈ (0, 0.5), R ∈ [0, 0.25], t ∈ [0, 1], x ∈ [0, 2]...
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0answers
34 views

Basis for a set of polynomials in Sage? [closed]

I have a large set of polynomials in the coordinates $x,y,z$ in Sage, (e.g. $x^5y-3x^2y^2+2xy^3+x^2yz-y^2z$). I want to know, for example, if $x^5y$ is in the span of my set. Is there a Sage command ...
1
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0answers
143 views

A Quaternions version of the Gauss Lucas theorem

Edit: My serious thanks to Christian, Todd and Yemon for their comments to the previous version. The derivative is the same as Todd says: $(az^{n})'=naz^{n-1}$. The polynomial is in the form of $\...
13
votes
1answer
417 views

Why sum of three squares of real polynomials is a sum of two squares?

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...
2
votes
1answer
279 views

Half spaces free of roots of a given polynomial

I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
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0answers
172 views

Enumerating a class of polynomials

How many equivalence classes of $\Bbb F_2[x,y]$ polynomials with $x$ degree $n_x$ and $y$ degree $n_y$ are there such that each $y^i$ coefficient (polynomial in $\Bbb Z[x]$) is distinct and $x^i$ ...
2
votes
1answer
59 views

Uniform Mahler Measure Lower Bound

I'm working on a problem in multiplicative ergodic theory, and Mahler measure has just made another appearance. I am looking for a uniform lower bound on Mahler measure over all polynomials of fixed ...
5
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0answers
79 views

Calculate Ramanujan's class invariant by using modular equation of degree $5$

Let $$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}=\frac{\pi}{2}{ _2F_1\bigg(\frac{1}{2},\frac{1}{2},1;k^2 \bigg)}$$ where $0<k<1$ Let $K, K′, L$ and $L′$ denote the ...
6
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1answer
483 views

Is this a semi algebraic set?

Put $$A=\{(a_{0},a_{1},\ldots,a_{n}) \in \mathbb{C}^{n+1}\mid p(z)=a_{0}+a_{1}z+\ldots a_{n}z^{n} \;\;\text{is a one-to one function on the unit disc} \{z\in \mathbb{C} \mid |z|\leq 1\}$$ Is $\{(...
8
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1answer
371 views

Polynomial approximations of curves

This is the 3D version of this question. The responses to that question contained a lot of complaints about fuzzy definition of the problem, so I made this new question very narrow and explicit. For ...
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0answers
46 views

Does cutting off the taylor expansion of e^x always give an irreducible polynomial? [duplicate]

I am talking of the polynomials: $P_n(x)$ = $1+x..+x^n/n!$ I've tested this for the first 10 values and it seems so. I know this might be random but I've got a hunch that there's something deeper ...
8
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1answer
211 views

$p | f(x)$ if and only if $p^k | x$.

Given a prime number $p$ and a positive integer $k$. Consider integer-valued polynomials $f$ satisfying the property that $p | f(x) \Leftrightarrow p^k | x$. Question. What is the smallest degree of ...
8
votes
1answer
73 views

Schur positivity on 2 letter alphabets implies Schur-positivity on n letters?

Suppose we have a symmetric polynomial $P$ in $n$ variables. We can partition this alphabet into sets with one or two letters, e.g. ${ {x_1}, {x_2, x_3}}. We can thus see $P$ as an element in $Q[x_1]...
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votes
1answer
156 views

How many the distinct linear factors of $f(x)-f(y)$ can be for f in Q[x]?

Let $f \in \mathbb{Q}[x]$. Let $S(f)$ denote the number of distinct linear factors of $f(x)-f(y)$. $S(f)$ is bounded by $\deg(f)$. Q1 Is $S(f)$ bounded by constant? Q2 Is it possible $S(f)&...
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0answers
50 views

Bounds on the positive roots of a bivariate polynomial

It is well known that various real root isolation methods are based on computing, first, the bounds on the values of the positive real roots of a polynomial equation. For the univariate case such ...
4
votes
1answer
373 views

Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$

Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$ Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$? I can show it when $n$ is a ...
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0answers
72 views

Question about radical ideal

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) \...
6
votes
1answer
223 views

On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have $$ \textrm{inf}(f(x)) > 0 \implies \textrm{inf}(f(x)) \geq \frac{3}{4} . $$ Could we generalize this (for ...
0
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0answers
33 views

Partial Fraction Decomposition based on Monomials

Given two positive integers $m$ and $k$, complex $a$ and the rational polynomial $$ q(z) = \frac{1}{z^{m+k} + a z^{m} + a z^{k} + 1}. $$ Is there a partial fraction expansion over the complex numbers, ...
0
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0answers
160 views

Warren's Theorem

At the end of page 12 in this document Noga Alon mentions Warren's Theorem on sign patterns: tau.ac.il/~nogaa/PDFS/tools1.pdf Does anyone know of an intuitive explanation of the proof of it ? Also, ...
3
votes
1answer
125 views

A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let $f_{...
4
votes
1answer
140 views

Irreducible monic polynomials

I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here. For instance, for the family of ...
9
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0answers
95 views

Factorisation in $\mathbb{N}[X]$?

Do we know an efficient algorithm to factorise in $\mathbb{N}[X]$ ? One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and to combine some factor to ...
5
votes
1answer
95 views

Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\...
3
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0answers
71 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...
4
votes
2answers
202 views

Invariant polynomials under diagonal action of the orthogonal group

Consider the diagonal action of the orthogonal group $O(n)$ on $\mathbb{R}^n\times\mathbb{R}^n$ defined as: $U\cdot (x,y) = (Ux,Uy)$ for $U\in O(n)$ and $x,y\in\mathbb{R}^n$. I am looking for a ...
0
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0answers
29 views

Does this system have a closed-form solution? $x_j = \left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $

I am interested in solving the following system of $n$ equations: $$x_j = \left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $$ for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, $0<\alpha<...
2
votes
0answers
72 views

Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
4
votes
0answers
48 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
340 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 ...
4
votes
1answer
179 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
68 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
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1answer
91 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
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0answers
103 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 \...
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0answers
62 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
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0answers
80 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
198 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
100 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
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0answers
116 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} \binom{d}{\alpha}...
3
votes
0answers
166 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
407 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
121 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) $$ p:S^2S^3\mathbb{C}...
2
votes
0answers
113 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
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0answers
33 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
105 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
182 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 $L^{\...