math.GM — General Mathematics

Galois groups of Taylor polynomials: prescriptions and arithmetic families

Contributed by Khai-Hoan Nguyen-Dang

Let $f_n(x)=\sum_{j=0}^n a_jx^j$ be the Taylor polynomials of a power series with nonzero rational coefficients. We study the proportions of degrees for which the Galois group over $\Q$ is symmetric, alternating, or neither. We prove that the set of degrees $n\geq3$ with symmetric group can be chosen arbitrarily, even for an entire function with positive rational coefficients whose Taylor polynomials have distinct negative real roots. More precisely, at each degree $n\geq3$ outside the chosen set, $f_n$ has exactly one rational root, and the group on the other roots is the full symmetric group. Thus the proportions need not converge; when they do, the proportion of other groups can be any number in $[0,1]$. These conclusions answer negatively two questions of Katz and Rivin. Moreover, for fixed $s\in\Q\setminus\Z$, we also give a condition on the prime divisors of rational multipliers of the coefficients $s/(s+j)$ that forces symmetric groups in density one. For the Taylor polynomials of $(1-x)^{-s}$, we determine the groups in all sufficiently large degrees. The number of exceptions to the symmetric group up to degree $X$ is $c_s\sqrt X+O_s(1)$, with an explicit constant $c_s\geq0$. We also determine the groups of the central-binomial Taylor polynomials in every degree and prove that every nonconstant Catalan Taylor polynomial has symmetric group. This proves Sun's Catalan group conjecture and his irreducible-reduction criterion for the monic reciprocals of the central-binomial polynomials.