## Abstract

Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of an N-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rank N for a function of a g-dimensional argument that generalize the classical functional Cauchy equation. It is shown that for N=2 the general analytic solution of these equations is determined by the Baker-Akhiezer function of an algebraic curve of genus 2. It is also shown that θ functions give solutions of a Cauchy equation of rank N for functions of a g-dimensional argument with N≤2^{g} in the case of a general g-dimensional Abelian variety and N≤g in the case of a Jacobian variety of an algebra curve of genus g. It is conjectured that a functional Cauchy equation of rank g for a function of a g-dimensional argument is characteristic for θ functions of a Jacobian variety of an algebraic curve of genus g, i.e., solves the Riemann-Schottky problem.

Original language | English |
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Pages (from-to) | 142-149 |

Number of pages | 8 |

Journal | Theoretical and Mathematical Physics |

Volume | 94 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1993 |

Externally published | Yes |