# 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
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### Stability analysis with minimal spectral norm

Let $A \in \mathbb{R}^{n \times n}$ with $$s(A) = \inf_{D \textrm{ is diagonal}} \| D^{-1} A D \|_2 > 1$$ Does there exists $m \in \mathbb{N}^n$ and $z \in \mathbb{C}$ with $|z| > 1$ such ...
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### How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?

Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
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### Testing polynomials irreducible over the integers

Let $f\in\operatorname{int}(\mathbb Z)$, the ring of integer-valued rational polynomials. Define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments....
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### Does linear independence imply algebraic independence for partitioned homogeneous polynomials?

Define a partitioned homogeneous polynomial of degree $d$ to be a polynomial in $$\mathbb Z[x_{11},\dots,x_{1n},\dots,x_{d1},\dots,x_{dn}]$$ with monomials from entries in (polynomials that are $d$-...
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### If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is

Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer. In a previous post I asked if $p_n(a,b)$ was a ...
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### Elementary proof for Hilbert's irreducibility theorem

I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding. I am only interested in the simple case where the ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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}\... 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 (... 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 ... 1answer 137 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 \... 0answers 196 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 $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\... 1answer 292 views ### A product of polynomials Let f(n)=1+x^n+x^{2n}+...+x^{n^2}. Let p(x) be 1+x+x^2+x^5+x^7+... where the exponents are the pentagonal numbers. Let a(n) be the sequence of integers such that the coefficients of the ... 1answer 47 views ### Lie-algebra-like relation for totally symmetric 4-tensors There are many totally symmetric real 4-tensors, T_{ijkl}, which satisfy the relation$$T_{ijmn}T_{mnkl} + T_{ikmn}T_{mnjl} + T_{ilmn}T_{mnjk} = c T_{ijkl}$$with some constant c. By the way ... 1answer 262 views ### Reason Coppersmith fails here? Take classic problem of finding P,Q in balanced semi-prime N=PQ. P has a binary expansion and so does Q. We can set the binary 0/1 variables to be x_1 through x_{\lceil\log P\rceil} and ... 0answers 50 views ### Method of Coppersmith optimal for multivariate? It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ... 1answer 431 views ### Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets Assume x is a variable belongs to \mathbb R \setminus \{ 0,-1,+1 \} and consider for all i, j \in \mathbb N,$$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$then for all n \in \mathbb N ... 0answers 55 views ### Relation between lifts of simple roots and lifts of idempotents (Henselian property) Let f:A\to B be a morphism of commutative rings. Given a monic \varphi\in A[x] write Z(A,\varphi) for the set of simple roots of \varphi in A. Consider the following properties of f:A\to B.... 0answers 208 views ### How localized can a polynomial be in the L1 norm? Let 0<s<2 be a parameter, \Omega = [-1,1], and \Omega_s\subset \Omega be a set of measure s. I would like to bound the following ratio from above:$$\sup_{p\in\mathcal{P}_n} \frac{\... 1answer 166 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 ... 1answer 364 views ### Difference between Chebyshev first and second degree iterative methods Consider linear equation$Au = f$. We want to solve it with iterative method (assuming$A$is good). First order iterative method is: $$u^{k+1} = u^k - \alpha_{k+1}(Au^k - f),$$ The second degree ... 1answer 114 views ### Divergence of a series related to Schinzel's hypothesis H The Series Consider the series identity $$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$ $$R(n) = \left\... 1answer 186 views ### Nonnegative coefficients of a product of polynomials Let P(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n]. Does there exist an algorithm to decide whether there is a nonzero polynomial Q(x_1,\dots,x_n)\in\mathbb{R}[x_1,\dots,x_n] such that the product ... 1answer 249 views ### Linear difference inequality It is well known how to find a solution for the following linear difference equation$$h_{m} = h_{m-1} + a \cdot h_{m-2}$$Finding the roots r_1 and r_2 of r^2 - r - a, we have that the ... 0answers 97 views ### On the sum \sum_{x=0}^{(p-1)/2}(\frac{x^{4n}+cx^{2n}+d}p) with p an odd prime Let p be an odd prime, and let n be a positive integer. For c,d\in\mathbb Z we define$$F_p^{(n)}(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^{4n}+cx^{2n}+d}p\right),$$where (\frac{\cdot}p) is ... 0answers 63 views ### Products of different cyclotomic polynomials Is there a classification of all products of different cyclotomic polynomials with non-negative coefficients? Clearly if the cyclotomic polynomials have only nonnegative coefficients, their products ... 0answers 657 views ### Which sets of roots of unity give a polynomial with nonnegative coefficients? The question in brief: When does a subset S of the complex nth roots of unity have the property that$$\prod_{\alpha\, \in \,S} (z-\alpha)$$gives a polynomial in \mathbb R[z] with ... 1answer 332 views ### Subsets E of \mathbb{F}_{p^k} with vanishing polynomial subset sums The following question arose in some discussions recently as a misunderstanding of another problem. Question: Which subsets E\subset \mathbb{F}_{p^k} satisfy the property that \sum\limits_{x\in E}... 1answer 524 views ### On a (possible?) equivalence of Bunyakovsky conjecture Dirichlet's Theorem on primes in arithmetical progressions is equivalent to the following statement: for all integer numbers a,b, where \gcd(a,b)=1, there exists at least one prime of the form an+... 1answer 170 views ### Products of Cyclotomic Polynomials with Nonnegative Coefficients I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients. Some ... 1answer 135 views ### On a special type of subring of \mathbb C[x_0,…,x_{q-1}] Let p,q be odd primes. Consider the polynomial ring \mathbb C[x_0,...,x_{q-1}]. For m=0,1,...,p-1, let$$\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)... 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 ... 1answer 91 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 ... 7answers 1k views ### On the polynomial$\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}$Let$n = 2m$be an even integer and let$F_n(X)$be the polynomial $$F_n(X):=\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}.$$ I observed (but cannot prove) that the polynomial$F_n$is always divisible by$...
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 $\lim_{h \to 0} o(h)/h =0$. ...