Summary: ### 519.4, 512.54
V. I. ARNOLD
Steklov Mathematical Institute
8, ul. Gubkina, Moscow 117966, Russia
HIGHER DIMENSIONAL CONTINUED FRACTIONS
Received September 3, 1998
The higherdimensional analogue of a continuous fraction is the polyhedral surface, bounding the convex hull of the
semigroup of the integer points in a simplicial cone of the euclidian space. The article describes some conjectures and
theorems, extending to such higherdimensional continouos fraction the Lagrange theorem on quadratic irrationals
and the Gauss--Kuzmin statistics.
To Jšurgen Moser admiringly
1. Classifications of graded algebras
While canoeing at the Onega lake and at the Svjatukha river in the summer of 1988, I became so
nostalgic for mathematics that decided one rainy day to study, following the example of the algebraists,
a problem having no relations to any serious mathematics.
This attempt to produce something useless ended with a failure. The present article is an account
of the theories emerging from the investigation of the following unmotivated definition.
Consider a commutative associative graded algebra over the field C (or over R).
Definition 1. The algebra is called Aalgebra if its Poincar'e series equals 1 + t + t 2 + : : : (that is, if