Vol 86, No 1 (2022)

We study the asymptotic properties of multiple orthogonal Hermite polynomialswhich are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of our asymptotic analysis is a four-term recurrence relation connecting the polynomials with adjacent numbers. We obtain asymptoticexpansions as the number of the polynomial and its variable grow consistently (the so-called Plancherel–Rotach type asymptotic formulae). Two techniques are used. The first is based on constructing expansions of basesof homogeneous difference equations, and the second on reducing difference equationsto pseudodifferential ones and using the theory of the Maslov canonical operator.The results of these approaches agree.

Let $G$ be a finite group of Lie type $F_4$ and $W$ the Weyl group of $G$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement of $T$ in its algebraic normalizer $N(G,T)$. In particular, we find all the maximal tori that have a complement in $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. We find the minimal orders of the lifts of the elements $w$ in $N(G,T)$.

An exponential sum is a linear combination of characters of the additive groupof $\mathbb C^n$. We regard $\mathbb{C}^n$ as an analogue of the torus$(\mathbb{C}\setminus0)^n$, exponential sums as analogues of Laurent polynomials,and exponential analytic sets ($\mathrm{EA}$-sets), that is, the sets of common zerosof finite systems of exponential sums, as analogues of algebraic subvarieties of the torus.Using these analogies, we define the intersection number of $\mathrm{EA}$-sets andapply the De Concini–Procesi algorithm to construct the ring of conditions of the correspondingintersection theory. To construct the intersection number and the ring of conditions, weassociate an algebraic subvariety of a multidimensional complex torus with every$\mathrm{EA}$-set and use the methods of tropical geometry. By computing the intersectionnumber of the divisors of arbitrary exponential sums $f_1,…,f_n$, we arrive at a formulafor the density of the $\mathrm{EA}$-set of common zeros of the perturbed system $f_i(z+w_i)$,where the perturbation $\{w_1,…,w_n\}$ belongs to a set of relatively full measurein $\mathbb{C}^{n\times n}$. This formula is analogous to the formula for the numberof common zeros of Laurent polynomials.