We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of 73 master integrals.

We give a detailed account of algorithms and applications provided with SINGULAR:PLURAL (we call it PLURAL for short). The poster is done in form of (big) one-page introduction to the capabilities of the system.

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We also address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

What is SINGULAR? SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR's main computational objects are polynomials, ideals and modules over a large variety of rings, including important non-commutative rings. SINGULAR features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root-finding, visualization, and many more related functionalities. What is new? Gröbner bases over ringsThe Gröbner basis routines were modified to work for polynomial rings with coefficients from a ring like Z or Z/m instead of coming from a field. Coefficient rings of the form Z/(2 n ) are important for the applications, for example in proving the arithmetic correctness of data paths in System-on-Chip modules:We start with a set of equations G j , j = 1, . . . , m given by polynomials f j ∈ Z[X], X a set of variables, which are of the form G j :For the variables rk ∈ X in this equation we assume r ( j) i = a (l) k for 1 ≤ l ≤ j and all i, k. We call the variables a ( j) i inputs and r ( j) i outputs of G j . For every proof goal, we obtain an additional polynomial g depending on a subset of variables {a 1 , . . . , a t } ⊂ X and need to check whether g(a 1 , . . . , a t ) = 0 mod 2 n for all solutions of the set of equations {G j }. Example 1 A k-bit comparator of operands a and b is modeled by the polynomialDenote the set of all solutions to {G j } as V ({G j }). Analogously let V (g) be the set of all roots of g. We create an equivalent variety subset problem V ({h i }) ⊂ V (g) where h i and g are polynomials over a single ring Z/2 N with appropriate N. This problem can be solved effectively using Gröbner basis techniques, such as normal form computations.This methods was used to verify a multiply-accumulate-unit (which shall compute a + b * cmod2 6 4) and was much faster than other methods.Additional to the definition of Gröbner basis over rings (a generating set of polynomials for the ideal, whose leading terms generate the leading term ideal) we need the notion of a strong Gröbner basis: a Gröbner basis, where the leading term of each element of the ideal is divisible by an element of the strong Gröbner basis.SINGULAR computes a strong Gröbner basis with a variant of the algorithm of Buchberger.180

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