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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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        1answer
        170 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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        votes
        1answer
        93 views

        Hyper-simultaneous equation [on hold]

        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 ...
        1
        vote
        2answers
        137 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{...
        1
        vote
        0answers
        45 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 ...
        0
        votes
        1answer
        86 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,...
        4
        votes
        1answer
        144 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 +...
        6
        votes
        0answers
        93 views

        Irreducibility of q-factorial plus 1

        Let $q$ be a formal variable and for every positive integer $n$ let $$[n]_q! = 1 (1 + q)(1 + q + q^2) \cdots (1 + q + \cdots + q^{n-1})$$ be the $q$-factorial. Is it true that $[n!]_q + 1$ is an ...
        3
        votes
        1answer
        107 views

        Representation of Subgraph Counts using Polynomial of Adjacency Matrix

        We consider a graph $G$ of size $d$ with adjacency matrix $A$, whose entries take value in $\{0,1\}$. We are interested in the number of a certain connected subgraph $S$ of size $k$ in $G$. For ...
        8
        votes
        1answer
        101 views

        Log-concavity of repeated convolution

        Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
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        17 views

        Is the Weak Popov Form of a matrix over a polynomial ring unique?

        The Weak Popov Form of a matrix is a type of normal form of the matrix. We can find some special properties by changing each matrix to this form by considering some elementary operations on the matrix....
        3
        votes
        1answer
        99 views

        Cipolla's Prime numbers function: Computing the coefficients of the polynomial

        In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited: [1]: M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti ...
        4
        votes
        1answer
        227 views

        Why do polynomials $x^n + 1 \bmod N$ close a shorter cycle when $n$ is even than when $n$ is odd?

        Polynomials $f(x) \bmod N$, where $f(x)$ is of integer coefficients and $N$ is a composite of two distinct primes $p, q$, form a cycle --- usually leaving a tail because the cycle tends to close not ...
        1
        vote
        1answer
        58 views

        Density of a set of numbers dividing a fixed number with polynomial exponent

        Fix a positive integer $a>1$ and let $f\in\mathbb{Z}[x]$ be a polynomial with positive leading coefficient. We define a set $S$ of positive integers, $$ S=\{n\in\mathbb{Z}^+:n\mid a^{f(n)}-1\}. $$ ...
        0
        votes
        0answers
        54 views

        Hermite polynomial [migrated]

        \begin{equation}h''(x) - 2xh'(x) = -2 \lambda h(x), x \in R \end{equation} where $\lambda$ is the eigenvalue. \begin{equation} G(t,x) = \sum_{n=0}^ {\infty} t^nh_n = e^{2tx-t^2} \end{equation} This ...
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        vote
        0answers
        88 views

        Fine tuning the growth rate of the degrees of polynomials

        Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$. Suppose that $$\...

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