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.

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Parametrize regions of positivity of a polynomial

I realize that this problem is extremely generic, so I am pessimistic that there may be concrete solutions, but let me try... Consider a multi-variate polynomial $P(x)$, is it possible to find ...
1 vote
1 answer
116 views

Largest part and length of a partition in play

If $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k)\vdash n$ is an integer partition of $n$ then $\lambda_1$ is its largest part and $k$ is its length, $\ell(\lambda)$. Define the statistic $...
3 votes
2 answers
139 views

Vanishing of a product of cyclotomic polynomials in characteristic 2

Let $n$ be a positive integer and $1\leq j\leq n$. Consider the following polynomial: $$p_{n,j}(x)=\frac{\prod\limits_{i=1}^{n+1}\frac{x^{i}+1}{x+1}}{\prod\limits_{i=1}^{j}\frac{x^{i}+1}{x+1}\prod\...
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1 vote
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Dimension of linear compositions of polynomials

Let $P_1\dots P_s$ be s given polynomials in $n$ variables of degree $d$. The set of polynomials $\sum_k Q_k P_k$ defines a vector space, where $Q_k$ are polynomials with degree not greater than $d_k$....
1 vote
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The asymptotic growth of codimension of range of polynomial differential equation on finite fields

Inspired by the seminal paper of Andre Weil on the number of solutions of equations on finite fields we would like to present the following question: Let $P(x,y), Q(x,y)$ be two polynomials of ...
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40 views

Gegenbauer polynomial relation with complex argument

Gegenbauer polynomials, $C_j^{\nu}(t)$, are defined to be the coefficient of $h^j$ in the expansion $(1-2ht +h^2)^{-\nu}$. It can be shown using [Higher Transcendental Functions, Vol 1, Harry Bateman, ...
6 votes
2 answers
333 views

Optimal polynomial approximation of rational function $\frac{1}{1-x}$

I've been working on the following polynomial approximation problem. I want to find the optimal Chebyshev approximation of the rational function $\frac{1}{1-x}$ on the real interval $x\in[-\rho, \rho]$...
3 votes
1 answer
95 views

Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]

Let $a>b>0$. Suppose we want to minimize $$ f(x)=(x-a)^2+(1/x-b)^2, $$ over $x>0$. Equating $f'(x)=0$ leads to the quartic equation $$ g(x)=x^4-ax^3+bx-1=0. \tag{1} $$ Question: Is the ...
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2 votes
1 answer
133 views

Asymptotic analysis of an expression involving a Fox's H function

One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...
1 vote
1 answer
59 views

Numerical methods for systems of trilinear polynomials

I have some large system of particular non-linear polynomial equations: each equation mentions at most three variables no variable appears with a degree larger than 1. I'm not an expert in this area ...
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6 votes
1 answer
424 views

Construction of a symmetric polynomial in the roots that acts like the discriminant

The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
5 votes
1 answer
403 views

A polynomial identity related to Catalan numbers

Let $F_n^{(k)}(x)= \sum_j {\binom{n+(k-1)j}{kj} x^j}$ and $G_n^{(k)}(x)= \sum_j {\binom{n+j}{kj} x^j}.$ I am interested in the coefficients ${a_{n,k,j}}$ such that $$G_n^{(k)}(x)=\sum_{j\geq0 }{a_{n,...
5 votes
0 answers
285 views

Decomposition of a determinant

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...
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2 votes
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70 views

How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
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Padovan polynomials?

Is there a name for the polynomials $$P_n(k,x,s)=\sum_j\binom{n-j}{(k-1)n-kj}s^jx^{(k-1)n-kj}$$ with generating function $$\sum_{n\ge0} P_n(k,x,s)z^n= \frac{1}{1-s^{k-2}xz^{k-1}-s^{k-1}z^k}$$ which ...
1 vote
1 answer
62 views

Finding minimax approximation of a permutation equivariant polynomial

Is there any known method to approximate a given permutation-equivariant smooth function $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ as multivariable polynomial function $p: \mathbb{R}^{n} \to \mathbb{R}^{...
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11 votes
1 answer
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And, yet, another evaluation to Catalan numbers

Construct the $n$-tuple Cartesian product of the ternary set $X_n=\{0,1,2\}\times\cdots\times\{0,1,2\}=\{0,1,2\}^n$. Define its subset $W_n$ according to the rule (here $y=(y_1,\dots,y_n)$ is made use ...
2 votes
1 answer
119 views

Adjacent reducible polynomials

Let $P[X_1, X_2, \ldots, X_N]$ be a reducible polynomial in $\mathbb{Z}[X_1, X_2, \ldots, X_N]$ such that $P[X_1, X_2, \ldots, X_N] + 1$ is also reducible. What (if anything) can we say about $P$? One ...
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3 votes
0 answers
107 views

Detecting symmetries in polynomials that lead to nice geometric properties

If we plot the single variable polynomial $p(x) = (x^2-1)^2$, it is easy to see that it has a nice property: all of its local minima are actually global minima. In particular, it has precisely two ...
3 votes
1 answer
188 views

Powers of $2$ up to $2^{m-1}$ from a polynomial of degree $m-1$

Let $T(n,k)$ be a triangle of coefficients such that $T(n,k)\geqslant0$ for $n>0$, $0<k\leqslant n$, $0$ otherwise. Also $$T(2n+1,1)=\frac{1}{2n+1}, T(2n,1)=0$$ $$T(n,k)=\frac{1}{n}(T(n-1,k-1)+(...
2 votes
1 answer
154 views

Matching polynomial, but $K_2$ is replaced by $K_3$. Have these been studied?

Given a simple graph $G=(V,E)$, we can consider matchings, $M\subseteq E$, where $M$ is a matching iff no vertex is shared between different edges. The number of edges in $M$ is denoted $|M|$. The ...
3 votes
1 answer
161 views

Divisibility relation with a specific sum of divisors

Let $\sigma(n)$ be the sum of the divisors of $n$. Is it always true that if $n$ is odd, that $$n\mid\sum_{k=1}^{\frac{n-1}{2}}k^2\sigma(k)\sigma(n-k)?$$ I have checked this up to $n=100$, and I ...
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2 votes
0 answers
61 views

Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is $$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$ ...
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11 votes
1 answer
289 views

Tarski-Seidenberg for strict inequalities and bounded quantification

This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in ...
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3 votes
0 answers
162 views

Degree four polynomials with no real roots

Consider a degree four polynomial $$ f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x] $$ with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...
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7 votes
1 answer
251 views

Status of the fundamental theorem of algebra for the locale of real numbers

In constructive mathematics without any choice at all, it is well known that the fundamental theorem of algebra cannot be proven for the Dedekind real numbers. On the other hand, Wim Ruitenberg showed ...
4 votes
1 answer
220 views

Divisibility of special polynomials

Problem: classify all pairs $(k,P)$ such that $P(x)$ divides $$x^kP(x+1)+(x-1)^kP(x-1),$$ where $k\ge4$ is an integer, and $P$ a nonconstant monic polynomial with rational coefficients. I have found ...
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3 votes
0 answers
298 views

Is this FFT algorithm known?

Recently I've been thinking about alternatives to the usual Cooley-Tukey FFT for multiplying polynomials. I think I've come up with a pretty nifty algorithm for multiplying polynomials. So my question ...
1 vote
0 answers
247 views

Reductions of a system of equations at various primes

Let $f_1, \dots, f_n$ be a finite set of polynomials in the polynomial ring $Z[x_1, \dots, x_m]$. At a prime $p$, let $N_p$ be the number of solutions $x=(x_1, \dots, x_m)\in (\mathbb{Z}/p\mathbb{Z})^...
-1 votes
1 answer
72 views

finding positive roots for a polynomial [closed]

I have a polynomial, and I want to get the conditions for the number of positive roots What are the different methods out there to determine these conditions? this is the polynomial: f(g)=A1g^5 + A2g^...
7 votes
2 answers
337 views

A sequence of polynomials related to Catalan numbers

The sequence of polynomials $$P_n=\sum_{k=0}^{\lfloor(2n-1)/3\rfloor} \frac{(2n-2k-1)!(2n-2k-2)!}{k!(n-k)!(n-k-1)!(2n-3k-1)!}x^k$$ satisfies apparently the identities $$0=\sum_{j=0}^nP_{n-j}(P_j-(-x)^...
0 votes
0 answers
50 views

Probability of polynomials products to be bounded by a given bound

I am given a quotient ring $R=\frac{\mathbb{Z}[x]}{\left< x^n +t\right>}$ for $t\in\mathbb{Z}$, and two polynomials from $R$, $A$,$B$ and let $C$ to be there product. Defining the norms $$\Vert ...
16 votes
2 answers
913 views

Strengthening Ax-Grothendieck

The question was cross-posted from Math.SE: https://math.stackexchange.com/questions/4566017/strengthening-ax-grothendieck The question is simple. The Ax-Grothendieck theorem says a polynomial map $p\...
0 votes
0 answers
143 views

Continuous dependence of the (infinite) roots of a polynomial on its coefficients

I'm trying to show the continuous dependence of the roots of a polynomial on its coefficients when the root number can be infinite (e.g., $x-y$). I don't know much about algebraic geometry but after I ...
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8 votes
1 answer
343 views

Conditions for a power of a polynomial to have no negative coefficients

Consider a polynomial in one variable $p(x)$ with $p(0)>0$, and that is not a polynomial in $x^m$ for any $m>1$ (that is, the $gcd$ of the exponents appearing in $p(x)$ is 1). I would like to ...
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3 votes
0 answers
61 views

Monge–Ampère equation with polynomials

Let $P$ be a real polynomial of $n$ variables. Does the Monge–Ampère equation $$ \det D^2 Q=P $$ always have a (global) real polynomial solution $Q$? I think this should be something standard, but I ...
25 votes
1 answer
546 views

Does there exist a polynomial 𝑃(𝑥,𝑦) which detects all non-squares?

I asked this question in MathStackExchange back in April, and it received more than 30 upvotes, but no answer was offered even after a bounty. I am reposting it here in hopes that someone can answer ...
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1 vote
1 answer
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Can $P(z)$ have a divisor in a given congruence class?

In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...
2 votes
1 answer
203 views

Dimension of a general partial derivative of a linear subspace of polynomials

Let $U \subseteq \mathbb{C}[x_1,\dots, x_n]_d$ be a linear subspace of the vector space of homogeneous degree-$d$ polynomials (including zero). I would like a proof or counterexample of the claim that ...
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5 votes
0 answers
131 views

Higher Cardano formulae in terms of $\Theta$

Consider a polynomial in one variable with complex coefficient $$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
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1 vote
0 answers
48 views

Approximating a strictly increasing non-negative function on a non-negative domain by polynomials with non-negative coefficients

Let $f:[0,2]\rightarrow [0,\infty)$ be a strictly increasing smooth function. The Weierstrass approximation theorem says that we can uniformly approximate $f$ by polynomials. But my concern is ...
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2 votes
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75 views

Symmetric polynomial constructed from symmetric group

Let $n$ be a positive integer, $S_n$ be the symmetric group. For a permutation $p=[p_1,\dots,p_n]\in S_n$, define $x^p := x_1^{p_1}\cdots x_n^{p_n}$. It can be seen that the following polynomial is ...
3 votes
0 answers
63 views

Non-tree models of Lagrange inversion polynomials

The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...
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3 votes
2 answers
216 views

Inequality for Gaussian polynomials III

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two ...
6 votes
0 answers
199 views

Cardinality of a polynomial image $\pmod{p^n}$

Given $P(x) \in \mathbb{Z}[x]$ a polynomial, $n$ a positive integer and $p$ a prime, there is a result that relates $|\text{Im } P \pmod{p^n}|$ with $|\text{Im } P \pmod{p^{n+1}}|$ perhaps in terms of ...
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4 votes
0 answers
127 views

Nullstellensatz with nilpotents and $I=J(V(I))$

Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$ Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0. Let $f$ be a polynomial which is zero ...
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0 votes
0 answers
26 views

Specific analytical solution to a multivariate equation

I've encountered the following problem during my research, any help would be highly appreciated. Let a following multivariate equation be given: $F(x,P,I) = 0$, where $x$ is a variable and $P,I$ are ...
1 vote
0 answers
34 views

Lifting module homomorphisms imposing conditions on characteristic polynomials

Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
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1 vote
0 answers
60 views

A nonstandard polynomial interpolation problem

I thought I know the properties of polynomials quite thoroughly. But... I am looking for a recipe to construct, for any given finite set $S$ whose elements are sets of $N$ real vectors of length $n$ ...
0 votes
0 answers
87 views

Representing an $m$ dimensional quadratic polynomial as a polynomial on $\mathbb F_{q^m}$

We can represent $\mathbb{F}_{q^m}$ as $\mathbb{F}_q[\alpha]$ where $\alpha$ is root of an irreducible $m$-degree polynomial on $\mathbb{F}_q$. By sending $\sum_{i=0}^{m-1} c_i\alpha^i \mapsto (c_{m-1}...
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