# 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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### Parametrize regions of positivity of a polynomial

I realize that this problem is extremely generic, so I am pessimistic that there may be concrete solutions, but let me try... Consider a multi-variate polynomial $P(x)$, is it possible to find ...
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### And, yet, another evaluation to Catalan numbers

Construct the $n$-tuple Cartesian product of the ternary set $X_n=\{0,1,2\}\times\cdots\times\{0,1,2\}=\{0,1,2\}^n$. Define its subset $W_n$ according to the rule (here $y=(y_1,\dots,y_n)$ is made use ...
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Let $P[X_1, X_2, \ldots, X_N]$ be a reducible polynomial in $\mathbb{Z}[X_1, X_2, \ldots, X_N]$ such that $P[X_1, X_2, \ldots, X_N] + 1$ is also reducible. What (if anything) can we say about $P$? One ...
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### Detecting symmetries in polynomials that lead to nice geometric properties

If we plot the single variable polynomial $p(x) = (x^2-1)^2$, it is easy to see that it has a nice property: all of its local minima are actually global minima. In particular, it has precisely two ...
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### Specific analytical solution to a multivariate equation

I've encountered the following problem during my research, any help would be highly appreciated. Let a following multivariate equation be given: $F(x,P,I) = 0$, where $x$ is a variable and $P,I$ are ...
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### Lifting module homomorphisms imposing conditions on characteristic polynomials

Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
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### A nonstandard polynomial interpolation problem

I thought I know the properties of polynomials quite thoroughly. But... I am looking for a recipe to construct, for any given finite set $S$ whose elements are sets of $N$ real vectors of length $n$ ...
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### Representing an $m$ dimensional quadratic polynomial as a polynomial on $\mathbb F_{q^m}$
We can represent $\mathbb{F}_{q^m}$ as $\mathbb{F}_q[\alpha]$ where $\alpha$ is root of an irreducible $m$-degree polynomial on $\mathbb{F}_q$. By sending \$\sum_{i=0}^{m-1} c_i\alpha^i \mapsto (c_{m-1}...