VECTOR ADDITION THEOREMS AND BAKER-AKHIEZER FUNCTIONS

Functional equations that arise naturally in various problems of modem mathematical physics are discussed. We introduce the concepts of an N -dimensional addition theorem for functions of a scalar argument and C auchy equations of rank N for a function of a g-dimensional argument t hat generalize the classical functional Cauchy equation. It is shown t hat for N=2 the general analytic solution of these equations is determ ined by the Baker-Akhiezer function of an algebraic curve of genus 2. It is also shown that theta functions give solutions of a Cauchy equat ion of rank N for functions of a g-dimensional argument with N less-th an-or-equal-to 2g in the case of a general g-dimensional Abelian varie ty and N less-than-or-equal-to g in the case of a Jacobian variety of an algebra curve of genus g. It is conjectured that a functional Cauch y equation of rank g for a function of a g-dimensional argument is cha racteristic for theta functions of a Jacobian variety of an algebraic curve of genus g, i.e., solves the Riemann-Schottky problem.