On generating series of finitely presented operads (1202.5170v4)

Abstract: Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.

• The paper shows that the generating series for non-symmetric operads with a finite Gr"obner basis are algebraic functions.
• It demonstrates that the exponential generating series for shuffle regular symmetric operads are differential algebraic.
• New algorithms are introduced for efficiently computing operadic generating series that simplify to rational forms under certain conditions.

Overview of the Paper "On Generating Series of Finitely Presented Operads"

This paper explores the mathematical framework of operads, focusing specifically on the generating functions that describe the dimensions of their graded components. The primary objective is to provide a comprehensive understanding of these generating functions within the context of finitely presented operads endowed with a finite Gr\"obner basis.

Operads, as algebraic structures, provide a potent way to encode operations with multiple inputs and are instrumental in various areas of mathematics, including algebra, topology, and mathematical physics. The authors, Anton Khoroshkin and Dmitri Piontkovski, examine both non-symmetric and symmetric operads, offering significant results about the algebraic nature of their generating series.

Main Results

The authors articulate their findings through a series of theoretical results supported by examples and algorithmic approaches. The key results are:

1. Algebraic Nature of Generating Series for Non-Symmetric Operads: For non-symmetric operads equipped with a finite Gr\"obner basis, the ordinary generating series is shown to be an algebraic function. This implies that these series satisfy a polynomial equation with coefficients that are formal power series over the base field.
2. Differential Algebraic Generating Series for Shuffle Regular Symmetric Operads: In the symmetric context, particularly for shuffle regular operads with a finite Gr\"obner basis, it is proved that the exponential generating series is differential algebraic. This indicates the existence of a non-trivial differential equation encapsulating these series.
3. Conditions for Rational Generating Series: The authors further classify the conditions under which these generating series exhibit particularly simple forms. For shuffle regular operads, if the dimensional growth is bounded by an exponential function, the series simplify to rational functions.
4. Operational Algorithmicity: The paper provides new algorithms based on the theory of operadic Gr\"obner bases, enabling efficient computation of generating series for a wide class of operads. This includes describing differential equations governing these series and offering methods to explicitly solve them.

Implications and Future Directions

The theoretical advancements in this paper have several implications. Specifically, the results pave the way for deeper exploration into the algebraic nature of a broad class of operadic structures, enhancing our understanding of their combinatorial properties.

On a practical level, these findings can influence computational aspects in various domains such as algebraic geometry and theoretical computer science. The alignment of generating series with algebraic and differential algebraic functions suggests new computational techniques for deriving solutions to operadic problems.

The paper extends an invitation to further investigate operads not bounded by the current scope, suggesting that there may be equivalent results in more generalized settings or for related algebraic structures.

In anticipation of future work, the authors hint at the potential for these methods and results to be applied to even more complex structures, possibly those that arise in homotopy theory or algebraic topology, where operads play a crucial role.

Overall, Khoroshkin and Piontkovski's paper makes a significant contribution to the algebraic theory of operads, providing robust tools and insights that expand the capacity to analyze these intricate algebraic entities.