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Results tagged with polynomials
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user 5101
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.
12
votes
Polynomial with many integer but no other rational solutions?
This is a partial answer which highlights some of the subtleties of this question. I will use algebraic geometry language as this is the correct set up for such questions.
First, this question is only …
- 21.4k
11
votes
Proofs of the Chevalley-Warning Theorem
One of the best results in this vein which I know of is due to Hélène Esnault, and appears in the paper:
Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Inven …
- 21.4k
2
votes
Bounding the number of polynomials whose Galois group is a subgroup of the alternating group
The set
$$\{ \mathbf{a} =(a_1, \ldots, a_n) \in \mathbb{Z}^n : ||\mathbf{a}|| \leq B, \Delta(a_1,\ldots,a_n) \text{ is a square} \}$$
is a so-called thin set in the sense of Serre.
Serre studies the …
- 21.4k
7
votes
Random Diophantine polynomials: Percent solvable?
This implies that $100\%$ of polynomials of degree $d$ are irreducible over $\mathbb{Q}$, for fixed $d > 1$. … In particular $0\%$ of such polynomials have a rational root, hence $0\%$ have an integer root. …
- 21.4k
4
votes
Representing $x^6-4$ as a sum of two squares
This problem looks tricky. I'd recommend you look up "Châtelet surfaces"; these are a slightly easier case when one has a polynomial of degree $4$ instead of a polynomial of degree $6$, but a lot of t …
- 21.4k