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Shai Rosenberg

Publications and source records attributed to Shai Rosenberg.

3 recordsLinked to original sources
arXiv2024

Polynomial compositions with large monodromy groups and applications to arithmetic dynamics

For a composition $f=f_1\circ\cdots \circ f_r$ of polynomials $f_i\in \mathbb Q[x]$ of degrees $d_i\geq 5$ with alternating or symmetric monodromy group, we show that the monodromy group of $f$ contains the iterated wreath product $A_{d_r}\wr \cdots\wr A_{d_1}$. A similar property holds more generally for polynomials that do not factor through $x^d$ or Chebyshev. We derive consequences to arithmetic dynamics regarding arboreal representations, and forward and backward orbits of such $f$. In particular, given an orbit $(a_n)_{n=0}^\infty$ of $f$ as above, we show that for "almost all" $a\in \mathbb Z$, the set of primes $p$ for which some $a_n$ is congruent to $a$ mod $p$ is "small".

math.NT↗
arXiv2015

Square-free values of multivariate polynomials over function fields in linear sparse sets

Let f be a square-free polynomial in Fq[t][x] where Fq is a field of q elements. We view f as a polynomial in the variable x with coefficients in the ring Fq[t]. We study squarefree values of f in sparse subsets of Fq[t] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial f in Z[x] takes infinitely many square-free values. Let c(t) be a polynomial in Fq[t] of degree less than m, and let k < m be coprime to q. A consequence of the main result we show, is that if q is sufficiently large with respect to m and the degrees of f in t and x, then there exist $β_1,β_2$ in Fq such that $f(t,c(t)+β_1t^k+β_2)$ is square-free. Moreover, as q tends to infinity, the last is true for almost all $β_1$ and $β_2$ in Fq. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.

math.NT↗