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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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86 views

An idea related to the conjecture of Bunyakovsky [on hold]

Obviously a reducible polynomial $f\in\mathbb Z[X]$ only can have a few primes $p\in f(\mathbb Z)$, if any. Irreducible polynomials with a fixed prime divisor can at most have one prime $p\in f(\...
1
vote
2answers
170 views

Function on two variables that restricts to a polynomial

Lets say that I have a function $F(x,y)$ that is defined on nonnegative integers (or at least those are the values I care about) and is symmetric, so that $F(x,y)=F(y,x)$. Moreover, I know that for ...
3
votes
0answers
48 views

Biggest Cartesian Product Included in a Real Plane Curve

Suppose an irreducible smooth $p \in \mathbb{R}[x_1,x_2]$ is given, and we would like to find finite sets $S_1 , S_2 \subset \mathbb{R}$ such that $p(S_1 \times S_2)=0$ and $|S_1 \times S_2|$ is as ...
3
votes
1answer
187 views

A certain generalisation of the golden ratio

Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$ We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
2
votes
1answer
156 views

Integer valued polynomials over several variables

For simplicity this is about polynomials in just two variables. Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials $X^iY^j$ and therefore as a sum of polynomials $p_{ij}\...
5
votes
1answer
136 views

Deg $n$ integral polynomial $P(x)$ with $n+1$ integer solutions to $0\leq P\leq d$

Let $d\in\mathbf{N}$ be as follows: there exists a polynomial $P(x)$ with degree $n>1$ and integer coefficients, such that $P$ has $n+1$ integer solutions to \begin{equation*} 0\leq P(x) \leq d \...
2
votes
1answer
47 views

Lie-algebra-like relation for totally symmetric 4-tensors

There are many totally symmetric real 4-tensors, $T_{ijkl}$, which satisfy the relation $$T_{ijmn}T_{mnkl} + T_{ikmn}T_{mnjl} + T_{ilmn}T_{mnjk} = c T_{ijkl}$$ with some constant $c$. By the way ...
1
vote
0answers
54 views

Relation between lifts of simple roots and lifts of idempotents (Henselian property)

Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
1
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1answer
163 views

Intersection Solutions of four nonlinear equations

I have four nonlinear equations I want to find the points of intersection of these equations, and I used the software Mathematica, unfortunately after many hours of waiting it does not give me any ...
0
votes
1answer
262 views

Reason Coppersmith fails here?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. $P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and ...
9
votes
1answer
185 views

Nonnegative coefficients of a product of polynomials

Let $P(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$. Does there exist an algorithm to decide whether there is a nonzero polynomial $Q(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]$ such that the product $...
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0answers
92 views

On the sum $\sum_{x=0}^{(p-1)/2}(\frac{x^{4n}+cx^{2n}+d}p) $ with $p$ an odd prime

Let $p$ be an odd prime, and let $n$ be a positive integer. For $c,d\in\mathbb Z$ we define $$F_p^{(n)}(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^{4n}+cx^{2n}+d}p\right),$$ where $(\frac{\cdot}p)$ is ...
3
votes
1answer
114 views

Divergence of a series related to Schinzel's hypothesis H

The Series Consider the series identity $$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$ $$R(n) = \left\...
0
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0answers
63 views

Products of different cyclotomic polynomials

Is there a classification of all products of different cyclotomic polynomials with non-negative coefficients? Clearly if the cyclotomic polynomials have only nonnegative coefficients, their products ...
8
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0answers
207 views

How localized can a polynomial be in the L1 norm?

Let $0<s<2$ be a parameter, $\Omega = [-1,1]$, and $\Omega_s\subset \Omega$ be a set of measure $s$. I would like to bound the following ratio from above: $$\sup_{p\in\mathcal{P}_n} \frac{\...
3
votes
1answer
135 views

On a special type of subring of $\mathbb C[x_0,…,x_{q-1}]$

Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let $$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
11
votes
1answer
169 views

Products of Cyclotomic Polynomials with Nonnegative Coefficients

I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients. Some ...
1
vote
1answer
90 views

System of polynomial equations with a known root

I have 5 polynomial equations for 5 variables and I know that the set of roots is finite. All coefficients are integers. Ultimately I'd like to find all roots but finding the Groebner basis is ...
6
votes
0answers
74 views

An Ehrhart positivity question related to Schur polynomials

Consider the Schur polynomial $s_\lambda(x_1,\dotsc,x_k)$. It is easy to see from the hook-content formula for counting the number of semi-standard tableaux, that the function $$ n \to s_{n \lambda}(1,...
4
votes
1answer
114 views

complex polynomial

Let $N$ be a big integer number and consider the equation : $$ x^{N} + a_{N-1} x^{N-1} + ...+a_{1} x + o(\frac{1}{N})=0,$$ where $o(h)$ is by definition a term such that $\lim_{h \to 0} o(h)/h =0$. ...
3
votes
1answer
106 views

Marsden's Identity and B-splines

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following ...
6
votes
0answers
152 views

Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers

Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
2
votes
1answer
97 views

How do you quickly determine which coefficients are greater than zero when multiplying two univariate positive polynomials?

Suppose that I have two polynomials with a degree of $n$, $A(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ and $B(x) = b_nx^n + b_{n-1}x^{n-1} + ... + b_0$ and the coefficients of these polynomials are ...
4
votes
1answer
332 views

Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums

The following question arose in some discussions recently as a misunderstanding of another problem. Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
0
votes
0answers
105 views

Does Coppersmith technique suffice to factor?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. Is there evidence that no extension of Coppersmith technique will accomplish factoring $N=PQ$ in polynomial time? Technically I am ...
2
votes
0answers
41 views

Bernstein bound on the number of roots of a rectangular multivariable polynomial systems

I would like to know what Bernstein's bound is on multivariate polynomial systems where there are more equations than unknowns. I have a "generic" zero-dimensional multivariable polynomial system with ...
2
votes
0answers
195 views

Algebraically independent vectors in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\...
0
votes
0answers
104 views

Probability of degree $0$ gcd between every pair of random homogeneous polynomials shifted by random primes?

Take $n,d,B\in\mathbb Z_{>0}$ with $d<n$ and denote $\mathcal M_{n,d}$ to be set of all total degree $d$ monomials in $n$ variables $x_1,\dots,x_n$ with degree $\leq1$ in each variable (...
1
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0answers
50 views

Method of Coppersmith optimal for multivariate?

It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
3
votes
0answers
67 views

Independence of number fields generated by roots of Littlewood polynomials

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
6
votes
1answer
218 views

$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer

Does anyone know if the following problem has ever been studied? Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$ where $n$ is a ...
1
vote
0answers
55 views

About Kazhdan Lusztig polynomial evaluating at q=1

Given $w\le w'$ (in Bruhat ordering), does $P_{x,w}(1)\le P_{x,w'}(1)$ (in usual ordering of $\mathbb{R}$), where $P_{x,w}(q)$ is the Kazhdan Lusztig polynomial?
0
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0answers
55 views

$A,B,w \in \frac{\mathbb{Q}[t]}{(t^2-1)}[x,y]$: $\operatorname{Jac}(A,B)=1$, $\operatorname{Jac}(A,w)=0$, $w \notin \frac{\mathbb{Q}[t]}{(t^2-1)}[A]$

Let $R=\frac{\mathbb{Q}[t]}{(t^2-1)}$; trivially, $R$ is not an integral domain, since $(\overline{t-1})(\overline{t+1})=\overline{t^2-1}=\overline{0}$. Is it possible to find $A,B,w \in R[x,y]$ ...
1
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0answers
124 views

Prime generating polynomials

Continuation to this previous question. According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
9
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0answers
164 views

Semi-primes represented by quadratic polynomials

According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
3
votes
0answers
120 views

Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?

Let $R \in \{\mathbb{C}[t^2,t^3], \mathbb{Z}[\sqrt{5}]\}$, and let $A=A(x,y) \in R[x,y]$ with $\deg(A) \geq 1$ (total degree). I wish to prove or find a counterexample to the following claim: If ...
3
votes
1answer
227 views

Positive real root separation (v2)

(This is a follow-up question to Positive real root separation) Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,...
7
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0answers
205 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 ...
4
votes
1answer
230 views

Positive real root separation

Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal)....
2
votes
0answers
246 views

Asymptotics of Littlewood polynomials

Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients. He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether ...
3
votes
1answer
113 views

Roots of anti-palindromic polynomial if coefficients are odd.

This is in continuation of the question asked in this earlier post here. Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?
5
votes
1answer
200 views

$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied: (1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$. (2) ...
1
vote
1answer
111 views

Reference request for anti-palindromic polynomials.

I have come across a lot of papers that are written about the palindromic polynomials, however, I am recently interested in polynomials satisfying $$f(-x) = x^nf(1/x)$$ for $n\geq 1$ and for all $x\...
1
vote
1answer
132 views

For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
4
votes
1answer
255 views

A Conjecture about the integral related to Chebyshev polynomial

I am interested in the following integral related to the Chebyshev polynomials $$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$ where $n,m\in \mathbb{Z}^+.$ It is easy to see ...
1
vote
1answer
111 views

For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
3
votes
1answer
227 views

Orthogonal basis of polynomials?

Let us define the basis of polynomials given by: $$ \begin{array}\ P_0=1, \\ P_1=x, \\ P_2=x(x-1), \\ P_3=x(x-1)(x-2), \\ P_4=x(x-1)(x-2)(x-3), \ldots\\ \end{array} $$ I would like to know if this ...
23
votes
2answers
392 views

Does every positive continuous function have a non-negative interpolating polynomial of every degree?

Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
2
votes
1answer
116 views

$(x + y + z)…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$

$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity. The ...
0
votes
0answers
89 views

gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...