We study commutative associative polynomial operations A n ×A n → A n with unit on the affine space A n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

We describe homogeneous locally nilpotent derivations of the algebra of regular functions for a class of affine trinomial hypersurfaces. This class comprises all nonfactorial trinomial hypersurfaces. l in i in i

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В работе получено описание однородных локально нильпотентных дифференцирований алгебры регулярных функций некоторого класса триномиальных гиперповерхностей. Данный класс включает в себя все нефакториальные триномиальные гиперповерхности. Библиография: 16 названий.