Growth in varieties of multioperator algebras and Groebner bases in operads

Abstract

We discuss algorithmic approach to growth of the codimension sequences of varieties of multilinear algebras, or, equivalently, the sequences of the component dimensions of algebraic operads. The (exponentional) generating functions of such sequences are called codimension series of varieties, or generating series of operads. We show that in general there does not exist an algorithm to decide whether the growth exponent of a codimension sequence of a variety defined by given finite sets of operations and identities is equal to a given rational number. In particular, we solve negatively a recent conjecture by Bremner and Dotsenko by showing that the set generating series of binary quadratic operads with bounded number of generators is infinite. Then we recall algorithms which in many cases calculate the codimension series in the form of a defining algebraic or differential equation. For a more general class of varieties, these algorithms give upper and lower bounds for the codimensions in terms of generating functions and asymptotical bounds for the growth of codimensions. The upper bound (based on an operadic version of the Golod--Shafarevich theorem) is just a formal power series satisfying an algebraic equation defined effectively by the generators and the identities of the variety. The first stage of an algorithm for the lower bound is the construction of a Groebner basis of the operad. If the Groebner basis happens to be finite and satisfies mild restrictions, a recent theorem by the author and Anton Khoroshkin guarantees that the desired generating function is either algebraic or differential algebraic. We describe algorithms producing such equations. In the case of infinite Groebner basis, these algorithms applied to its finite subsets give lower bounds for the generating function of the codimension sequence.