Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams

This paper studies the global structure of algebraic curves defined by generalized unitarity cut of four-dimensional three-loop diagrams with eleven propagators. The global structure is a topological invariant that is characterized by the geometric genus of the algebraic curve. We use the Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with the help of techniques involving convex hull polytopes and numerical algebraic geometry. Some interesting properties of genus for arbitrary loop orders are also explored where computing the genus serves as an initial step for integral or integrand reduction of three-loop amplitudes via an algebraic geometric approach.Comment: 35pages, 10 figures, version appeared in JHE

Tensor decomposition and homotopy continuation

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties $X_1,\ldots,X_k\subset\mathbb{P}^N$ defined over $\mathbb{C}$. After computing ranks over $\mathbb{C}$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix multiplication with zeros. (26 pages, 1 figure

Numerical algebraic intersection using regeneration

In numerical algebraic geometry, algebraic sets are represented by witness sets. This paper presents an algorithm, based on the regeneration technique, that solves the following problem: given a witness set for a pure-dimensional algebraic set Z, along with a system of polynomial equations f: Z�C n, compute a numerical irreducible decomposition of V�Z�V f. An important special case is when Z�A B for irreducible sets A and B and f x,y�x y for x A, y B, in which case V is isomorphic to A�B. In this way, the current contribution is a generalization of existing diagonal intersection techniques. Another important special case is when Z�A C k, so that the projection of V dropping the last k coordinates consists of the points x A where there exists some y in a new set of variables such that f x,y�0. This arises in many contexts, such as finding the singularities of A, in which case f x,y can be a set of singularity conditions that involve new variables associated to the tangent space of A. The combining of multiple intersection scenarios into one common scheme brings new capabilities and organizational simplification to numerical algebraic geometry