Composite lacunary polynomials and the proof of a conjecture of Schinzel

On a class of lacunary almost Newman polynomials modulo p and density theorems

The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $ζ_β(z)$ of the $β$-shift, for $β> 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density th…

Explicit bounds for composite lacunary polynomials

Let $f, g, h\in \mathbb{C}\left[x\right]$ be non-constant complex polynomials satisfying $f(x)=g(h(x))$ and let $f$ be lacunary in the sense that it has at most $l$ non-constant terms. Zannier proved that there exists a function $B_1(l)$ on $\mathbb{N}$, depending only on $l$ and with the property that $h(x)$ can be written as the ratio of two polynomials having each at most $B_1(l)$ terms. Here, we give explicit estimates for this function or, more precicely, we prove that one may take for instance \[B_1(l)=(4l)^{(2l)^{(3l)^{l+1}}}.\] Moreover, in the case $l=2$, a better result is obtained u…

0,1系数缺项多项式的不可约性

Irreducibility of lacunary polynomials with 0,1 coefficients

本文证明了高次且项数较少的0,1-系数缺项多项式在均匀分布下以高概率不可约，并给出了其可约概率在极限下趋于0的严格数学证明。