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Results tagged with polynomials
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user 34180
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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Theorems proved using combinatorial nullstellensatz that have no other known proof
Alon's (or Alon and Tarsi's?) combinatorial nullstellensatz is a powerful algebraic tool with many applications in combinatorics and number theory. See this, this, this and this mathoverflow question. …
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10
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Proofs of the Chevalley-Warning Theorem
There is a bijection between functions from $\mathbb{F}_q^n$ to $\mathbb{F}_q$ and polynomials with degree in each variable at most $q-1$. … Since each $P_b$ vanishes everywhere except at $b$, by Lagrange interpolation, $P_b$'s span the whole space of polynomials/functions. …
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Polynomials of low degree that clone polynomials of higher degree
Here's a very general result that solves your problem.
Let $F$ be a field, and let $A = A_1 \times \dots \times A_n$ be a finite grid in $F^n$. A polynomial $P \in F[t_1, \dots, t_n]$ is called $A$- …
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10
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How to recognise that the polynomial method might work
Now, all these polynomials are degree $2$ homogenous polynomials in $n$ variables, and hence the number is bounded above by the dimension of this spaces, $n(n+1)/2$. … Peter Sziklai, Polynomials in finite geometries and Applications of Polynomials over Finite Fields.
Larry Guth, Polynomial Method in Combinatorics. …
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2
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Enumerating certain types of permutation polynomials
Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions:
$f(ax) = af(x)$ for all … We can see that this is an onto map where fiber of each perfect matching is a set of $(q-1)^{1+q+q^2}$ permutation polynomials. …
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Can Schwartz-Zippel be formulated for commutative rings instead of fields?
See Section 4 in "On Zeros of a Polynomial in a Finite Grid" to see how Schwartz-Zippel lemma and many similar results on zeros of polynomials work for arbitrary commutative rings as long as the "grid" …
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