Schinzel's factorization conjecture for sparse Laurent polynomials

Let FQ[x±1]F\in\mathbb{Q}[{\boldsymbol{x}}^{\pm1}]. A Laurent polynomial is cyclotomic if it is a unit times the composition of a univariate cyclotomic polynomial with a monomial. For each aZn{\boldsymbol{a}}\in\mathbb{Z}^n, write F(ta)F(t^{\boldsymbol{a}}) for the resulting Laurent polynomial in tt.

Schinzel's factorization conjecture. There is a finite set of matrices ΩZn×n\Omega\subset\mathbb{Z}^{n\times n} such that, for each aZn{\boldsymbol{a}}\in\mathbb{Z}^n, there are MΩM\in\Omega and bZn{\boldsymbol{b}}\in\mathbb{Z}^n with a=Mb{\boldsymbol{a}}=M{\boldsymbol{b}} such that if PP is an irreducible factor of F(xM)F({\boldsymbol{x}}^M), then P(tb)P(t^{\boldsymbol{b}}) is either a product of cyclotomic Laurent polynomials or an irreducible factor of F(ta)F(t^{\boldsymbol{a}}).

The conjecture seeks a uniform finite description of the non-cyclotomic factors arising under monomial specialization, while allowing cyclotomic factors as exceptional components. Its status is not resolved in the supplied text.

Sources & referencesView supporting material

Primary source

Francesco Amoroso and Martín Sombra, “Factorization of bivariate sparse polynomials”, arXiv:1710.11479 (2018).

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