In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progression are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is necessary in order to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor

A distance from free to dummy indices is defined. The distance is invariant with respect to both monoterm symmetries and bottom antisymmetry. Using the distance invariant, we present an index-replacement algorithm. We then develop two normalization algorithms. One is with respect to monoterm symmetries and has complexity lower than known algorithms; the other allows one to determine the equivalence

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case Genzmer and Paul (2011) and the quasi homogeneous case Genzmer and Paul (2016), we describe an algorithm which computes

We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ℓn where ℓ is the lacunary size of the input polynomial and n its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial

A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of unity. The theory of the so-called [Formula: see text]-extensions is supplemented by algorithms that support the construction of such difference rings automatically

Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL

We extend order-sorted unification by permitting regular expression sorts for variables and in the domains of function symbols. The obtained signature corresponds to a finite bottom-up unranked tree automaton. We prove that regular expression order-sorted (REOS) unification is of type infinitary and decidable. The unification problem presented by us generalizes some known problems, such as, e.g., order-sorted

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order

In this paper, we provide an algorithm to compute explicit rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its rational invariant algebraic curves. The method is based on the proper rational parametrization of these curves and the fact that by linear reparametrizations, we can find the rational solutions of the given system of ODEs. Moreover, if the