5
votes
1answer
268 views

Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero. Definition. A ...
6
votes
3answers
392 views

Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...
0
votes
0answers
85 views

Enumerating certain bounded polynomials with partition type property constraints

Let $p(x) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{d}x^{d} + \dots + a_{2d+\alpha}x^{2d+\alpha} \in \mathbb{Z}[x]$ where $\alpha \in \{0,1\}$ with the constraints $(1)$ $0 \leq ...
6
votes
1answer
400 views

Are roots of transcendental elements transcendental?

This looks extremely easy, but then again it's late at night... Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ if and only if ...
0
votes
1answer
96 views

Uniqueness of Hensel factors of a polynomial (invariant to change of “basepoint”)?

An important component of algorithms for factoring multivariate polynomials over a commutative ring $R$ is Hensel lifting. Here's a brief, concrete example to set the stage for my question: Let $f ...