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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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        2answers
        154 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 ...
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        votes
        0answers
        42 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 ...
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        votes
        1answer
        176 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
        154 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}\...
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        votes
        0answers
        59 views

        Racing Math equation [closed]

        I need a full equation to the below please: (positionscore == 1) x 25 + (positionscore >=2) x (positionscore <=7) x (26 - 2 x positionscore) + (positionscore >8) x (18 - positionscore) + ...
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        votes
        1answer
        133 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 \...
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        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 ...
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        vote
        0answers
        53 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 $f:A\to B$....
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        vote
        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 ...
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        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 ...
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        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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        votes
        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\...
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        votes
        0answers
        62 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 ...
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        votes
        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{\...

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