We show for a large class of entire functions, f, that after proper rescaling, on compact sets, the derivatives of f converge to cosine, in particular their roots become evenly spaced. This proves a conjecture of Farmer and Rhoades [Trans. Amer. Math. Soc., 357(9):3789–3811, 2005] and Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022] for our class of entire functions. A main ingredient of our proof is to show that high derivatives of high degree polynomials behave like Hermite polynomials, which we prove using the techniques from the newly developed field of finite free probability. This is joint work with Andrew Campbell and Sean O'Rourke.
Universality for roots of derivatives of entire functions
In this talk, we begin by considering the semiring of the natural numbers with truncated difference, N = (N, +, *, -, 0, 1). Its equational theory is not recursively axiomatizable, due to the fact that one can essentially encode in it the undecidability of Hilbert’s tenth problem. Our aim is to study finitely axiomatizable varieties of semirings with difference which include the one generated by N, and are as close to it as possible. To this end, we introduce the variety of almost natural semirings with difference, AN. We show that the subdirectly irreducible algebras in this variety coincide with the models of PA, a weak form of Peano Arithmetic without induction that is an axiomatic extension of Robinson Arithmetic. Models of PA are known to satisfy all Sigma_1 sentences true in N. This variety can be axiomatized by a few simple equations and enjoys several appealing universal-algebraic properties: it is connected to lattice-ordered rings, it is a discriminator variety, ideal determined and 0-regular; thus it admits an associated 0-assertional logic. Nevertheless, we will see that its connection to arithmetic still yields an undecidable equational theory. This talk is based on ongoing joint work with Guillermo Badia, Xavier Caicedo, and Carles Noguera.
Universality for roots of derivatives of entire functions