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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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        Irreducibility of root-height generating polynomial

        The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
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        0answers
        40 views

        A special case of the polynomial Bézout's identity: bounding the co-factors

        Let $F$ be a field of prime order $p$. Suppose that $f\in F[x]$ is a non-zero polynomial of degree $\deg f<p$. If $f$ does not have multiple roots, then there exist polynomials $P,Q\in F[x]$ such ...
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        50 views

        On the relation between elementary symmetric polynomial inequalities and $p$-norms

        Let $x, y \in \mathbb{R}^n_+$, and let $e_k$ denote the $k$th elementary symmetric polynomial. It is known (see e.g. [1]) that if \begin{align} e_1(x) &= e_1(y),\tag{1}\\ e_k (x) &\leq e_k (y)...
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        votes
        0answers
        86 views

        Upper bound over $[0,1] $ for strange family of polynomials

        Let $n$ integer $\geq 2,$ $x$ real, and $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \...
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        votes
        0answers
        38 views

        Degree bounds of polynomial factors

        Given a polynomial over the integers, we can use Dumas theorem to narrow down the degrees of potential factors. Are there any other theorem that give bounds on degrees of factors?
        2
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        0answers
        115 views

        Intuition behind the growth condition in the result of Griffin, Ono, Rolen and Zagier on Jensen Polynomials

        With great pleasure I read the recent paper of Griffin, Ono, Rolen and Zagier proving the surprising result that the Jensen polynomials $J^{d, n}_\alpha$ for a sequence $\alpha = \{\alpha(0), \alpha(1)...
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        45 views

        Division of bivariate polynomials

        The following theorem (lemma 4.2.18 on page 97) is proven in thesis "Computationally efficient Error-Correcting Codes and Holographic Proofs" by Daniel Alan Spielman: Let $E(X, Y)$ be a polynomial ...
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        0answers
        115 views

        Infinite Order Automorphisms of Planar Polynomials

        Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...
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        2answers
        718 views

        Roots of $x^n-x^{n-1}-\cdots-x-1$

        It is easy to see that $f(x)=x^n-x^{n-1}-\cdots-x-1$ has only one positive root $\alpha$ which lies in the interval $(1,2)$. But it is claimed that this root is a Pisot number, i.e., the other roots ...
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        92 views

        Solution existence theorems of polynomial system of equations [closed]

        I am not a math guy, just curious about the following question: For any multivariate real polynomial system of equations with degree of two, is there any standard way to show the existence or non-...
        5
        votes
        1answer
        218 views

        Is there a general geometric characterization for polynomials to be linearly dependent?

        Consider $P$ the complex projective plane, and fix a line $L$ in $P$ I had a conjecture, that prof. I. Dolgachev showed me how to prove, that $3$ quadratic polynomials depending on a variable $z \in ...
        2
        votes
        1answer
        62 views

        Intersection of quadratic equations with planted solutions?

        Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection? In general what can we say about intersection of $k$ quadratics? How many ...
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        0answers
        11 views

        Orthogonal fit in linearization [closed]

        I have a formula for orthogonal fitting(5th order in this case) that is used for linearization. It includes coefficients, which I would like to verify if they are correct but I can't find any ...
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        votes
        0answers
        60 views

        Non-isomorphic graphs with same Tutte polynomial [closed]

        I've been looking for some non-isomorphic graphs with the same Tutte polynomial. I'm aware of this thread and this thread, however my understanding of matroids is non-existent, and they are a bit ...
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        votes
        0answers
        76 views

        Intersection of an ideal and a subring

        Is there a Grobner basis method that can compute the intersection of an ideal $I$ of a polynomial ring $R$ and its subring $R^\prime$? For example, I have an ideal $I=(x+y+z^2,1+xyz+yz+xz)$ of $\...

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