 
Summary: ### 519.4, 512.54
\Gamma\Delta \Theta\Lambda
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 GaussKuzmin 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
