We consider the check of the involutive basis property in a polynomial context. In order to show that a finite generating set F of a polynomial ideal I is an involutive basis one must confirm two properties. Firstly, the set of leading terms of the elements of F has to be complete. Secondly, one has to prove that F is a Gröbner basis of I. The latter is the time critical part but can be accelerated by application of Buchberger's criteria including the many improvements found during the last two decades. Gerdt and Blinkov (Involutive Bases of Polynomial Ideals. Mathematics and Computers in Simulation 45, pp. 519-541, 1998) were the first who applied these criteria in involutive basis computations. We present criteria which are also transfered from the theory of Gröbner bases to involutive basis computations. We illustrate that our results exploit the Gröbner basis theory slightely more than those of Gerdt and Blinkov. Our criteria apply in all cases where those of Gerdt/Blinkov do, but we also present examples where our criteria are superior. Some of our criteria can be used also in algebras of solvable type, e. g., Weyl algebras or enveloping algebras of Lie algebras, in full analogy to the Gröbner basis case. We show that the application of criteria enforces the termination of the involutive basis algorithm independent of the prolongation selection strategy.
Motivated by arithmetic properties of partition numbers p(n), our goal is to find algorithmically a Ramanujan type identity of the form ∞ n=0 p(11n + 6)q n = R, where R is a polynomial in products of the form e α := ∞ n=1 (1 − q 11 α n ) with α = 0, 1, 2. To this end we multiply the left side by an appropriate factor such the result is a modular function for 0 (121) having only poles at infinity. It turns out that polynomials in the e α do not generate the full space of such functions, so we were led to modify our goal. More concretely, we give three different ways to construct the space of modular functions for 0 (121) having only poles at infinity. This in turn leads to three different representations of R not solely in terms of the e α but, for example, by using as generators also other functions like the modular invariant j.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.