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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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      Is the polynomial $g(t) = \sum_{q \text{ prime }, q\le p} t^{q-2}$ for a prime $p\ge 7$ separable? [on hold]

      Let $a_0 + a_1 x^1 + a_2 x^2 + \cdots + a_n x^n$ be a polynomial in $\mathbb{Z}[x]$ such that $a_0 \neq 0$ and $|a_i| \le 1$ for $i=0,1,\cdots,n$. Then my conjecture is that this polynomial is ...
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      0answers
      36 views

      A common zero for a homogeneous polynomial and its gradient implies the existence of a common non-constant factor

      Good morning, I've recently came across the following question: let $n\ge 2$ and $P\in \mathbb{R}[x_1,\dots, x_n]$ be an homogeneous polynomial of degree $d$. Let $v$ be a real projective common ...
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      75 views

      Approximating $3SAT$ by polynomials

      Given $3SAT$ formula $\phi$ in $n$ variables and a $\mu\in(0,1/2)$ what is the smallest degree of polynomial $f\in\mathbb Z[x_1,\dots,x_n]$ such that for an $(a_1,\dots,a_n)\in\{0,1\}^n$ if $\phi(a_1,\...
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      67 views

      Counting the number of separable polynomials of degree $n$, with a certain fixed constant

      I am trying to understand the following statement which appear in the following paper http://www-bcf.usc.edu/~fulman/LAApaper.pdf . Statement: The number of monic square-free polynomials of degree $n$...
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      0answers
      93 views

      Infinite products from the fake Laver tables-Now with no set theory

      We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have $2^{n}*_{n}x=x$, $x*_{n}1=x+1\mod 2^{n}$,...
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      Is this problem in $NP$?

      Where is Given two $n$ many algebraically independent homogeneous system of polynomials (hence zero-dimensional system) in $\mathbb Z[x_1,\dots,x_n]$ with degree $2$ with absolute value of ...
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      155 views

      Reconstructing almost known polynomial from a system of polynomials with common roots

      We have $n$ algebraically independent degree $2$ homogeneous system of polynomials with $\mathbb Z$ coefficients in $n$ variables with exactly $t$ primitive (gcd of coefficients is $1$) integer roots ...
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      33 views

      Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

      Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.
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      99 views

      If a and b are roots of polynomials P and Q, then what polynomials are a+b and ab a roots of? [closed]

      If $a \in \mathbb{R}$ and $b \in \mathbb{R}$ are roots of polynomials $P$ and $Q$ with rational coefficients, is there an algorithm / formula / process for finding polynomials with rational ...
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      1answer
      193 views

      Most points on a degree $p$ hypersurface?

      Let $p$ be a prime. Let $f \in \mathbb{F}_p[x_1, \ldots, x_n]$ be a homogenous polynomial of degree $p$. Can $f$ have more than $(1-p^{-1}+p^{-2}) p^n$ zeroes in $\mathbb{F}_p^n$? Basic observations: ...
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      1answer
      106 views

      Hyper-simultaneous equation [closed]

      I am new here and I just want to ask if the following system has a general solution: If a, b and c are given such that: $$ x + y + z = a $$ $$ x^2 + y^2 + z^2 = b $$ $$ x^8 + y^8 + z^8 = c $$ Is ...
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      2answers
      143 views

      Integral formula involving Legendre polynomial

      I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values. \begin{equation} \int_{-1}^{1}\sqrt{...
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      64 views

      Why does $p_{n}(i,1)=1$ so often where the polynomials $p_{n}$ are obtained from the classical Laver tables

      So I was doing some computer calculations with the classical Laver tables and I found polynomials $p_{n}(x,y)$ such that $p_{n}(i,1)=1$ for many $n$. The $n$-th classical Laver table is the unique ...
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      1answer
      103 views

      Symmetric polynomials in two sets of variables

      Suppose $f(x_1,...,x_m,y_1,...,y_n)$ is a polynomial with coefficients in some field which is invariant under permuting the $x$'s and the $y$'s. Then $f$ can be generated elementary functions $e_k(x_1,...
      6
      votes
      2answers
      257 views

      Maximize $L^p$ norm over sphere

      For $p \in \mathbb{R}$, consider the function $$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$ My goal is to maximize this function under the constraints that $$ \lambda_1^2 +...

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