Decomposability of multivariable polynomials

arXiv1008.4971MaRDI QIDQ6220431

Author name not available (Why is that?)

Publication date: 29 August 2010

Abstract: Let

K

be an algebrically closed field and let

n g e q 1

. If

P i n K [X] = K [X_{1}, l d o t s, X_{n}]

,

P e q 0

, we denote by

I (P)

the support of

P

, which is the finite subset of

m a t h b b N^{n}

such that

P = s u m_{i i n I (P)} a_{i} X^{i}

with

a_{i} i n K^{*}

. (If

i = (i_{1}, l d o t s, i_{n})

then

X^{i} : = X_{1}^{i_{1}} c d o t s X_{n}^{i_{n}}

.) We determine all finite, nonempty sets

I s b m a t h b b N^{n}

such that every

P i n K [X]

with

I (P) = I

is decomposable. We also consider the problem of finding all

I s b m a t h b b N^{n}

such that every

P i n K [X]

with

I (P) = I

is irreducible. We do not solve this problem, which is very unlikely to have a simple answer. We show however that the answer depends on the characteristic of

K

and we determine the nature of this dependence.

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