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Results tagged with polynomials
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user 1384
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.
9
votes
The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$
$D=D(f,g)$ is not particularly well-behaved, is it. For example it's not multiplicative in the variables: if $g=x^2+1$ (nothing special about this example, I'm just fixing ideas) then $D(f,g)$ is the …
- 41.4k
13
votes
On Polynomials dividing Exponentials
The source of the question is "Problems from the Book" by Andreescu and Dospinescu. I finally emailed Andreescu yesterday asking what was going on. He apologised---he says there's a typo in the book. …
- 41.4k
25
votes
English reference for a result of Kronecker?
If all the Galois conjugates of an algebraic integer $\alpha$ have absolute value at most 1, then the norm of this algebraic integer is a rational integer with absolute value at most 1. Hence either t …
- 41.4k
6
votes
On Polynomials dividing Exponentials
Ok so Gjergji deleted his answer because it was mistaken at a critical point, but I was lucky enough to see it, and using one of the ideas in the answer I think one can prove that $m$ is odd. This is …
- 41.4k
5
votes
Infinite collection of elements of a number field with very similar annihilating polynomials
The answer "is" that the smallest $r$ is what it is, and what it is could well depend on $\alpha$. Let me also raise the possibility that there might be no simple "formula" relating $r$ to $\alpha$. T …
- 41.4k
34
votes
Accepted
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
EDIT: Hendrik Lenstra emailed me a proof of Conjecture 2. I'll append it below. So Jagy's question is now solved.
OK so I think that Jagy wants to make the following conjecture:
CONJECTURE 1: an i …
- 41.4k
2
votes
Accepted
A question about the invariants of a finite group
of the $t^n$ term the factors must have degree less than $n$ in $t$ (consider the factorization in $\mathbb{Q}(y_1,y_2,\ldots,y_n)[t]$; note also that we can assume that the factors over $R$ are monic polynomials …
- 41.4k