The complexities of nonperturbative computations

The paper studies the behavior of equations of motions of Green’s functions under different running coupling constants in strongly coupled gauge field theories in terms of the Kolmogorov complexity

On the Complexity of Computing Galois Groups of Differential Equations

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions. Hrushovski first proposed an algorithm for computing the differential Galois group of a general linear differential equation. Recently, Feng approached finding a complexity bound of the algorithm, which is the degree bound of the polynomials used in the first step of the algorithm for finding a proto-Galois group. The bound given by Feng is quintuply exponential in the order n of the differential equation. The complexity, in the worst case, of computing a Gröbner basis is doubly exponential in the number of variables. Feng chose to represent the radical of the ideal generated by the defining equations of a proto-Galois group by its Gröbner basis. Hence, a double-exponential degree bound for computing Gröbner bases was involved when Feng derived the complexity bound of computing a proto-Galois group. Triangular decomposition provides an alternative method for representing the radical of an ideal. It represents the radical of an ideal by the triangular sets instead of its generators. The first step of Hrushovski\u27s algorithm is to find a proto-Galois group which can be used further to find the differential Galois group. So it is important to analyze the complexity for finding a proto-Galois group. We represent the radical of the ideal generated by the defining equations of a proto-Galois group using the triangular sets instead of the generating sets. We apply Szántó\u27s modified Wu-Ritt type decomposition algorithm and make use of the numerical bound for Szántó\u27s algorithm to adapt to the complexity analysis of Hrushovski\u27s algorithm. We present a triple exponential degree upper bound for finding a proto-Galois group in the first step of Hrushovski\u27s algorithm

Elimination for Systems of Algebraic Differential Equations

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data of the system of differential equations through the theory of differential kernels to provide a new upper bound for proving the consistency of the system. We then prove a new upper bound for the effective differential Nullstellensatz, which determines a sufficient number of times to differentiate the original system in order to prove its inconsistency. Finally, we study the Rosenfeld-Gröbner algorithm, which approaches differential elimination by decomposing the given system of differential equations into simpler systems. We analyze the complexity of the Rosenfeld-Gröbner algorithm by computing an upper bound for the orders of the derivatives in all intermediate steps and in the output of the algorithm

Lecture as part of Graduate Center Series For Kolchin Seminar in Differential Algebra, 2005–6

I’ll discuss some open problems related to differential algebra for which experiments in computations may be helpful. Examples showing how to set up such experiments will be given. There are two obvious points I want to make: 1. Computations become necessary when we want to compute something, like solving a system of (differential) equations, or finding some examples or counterexamples. 2. In turns, computational consideration of effectiveness and efficiency may itself lead to mathematical problems, such as complexity, enumeration of vector space basis, encoding of input and output, etc. Some of these problems are of interest in its own right. Advice: You can write a dissertation or paper on the new problems if you cannot solve the original computation problem! 2 Gaussian Eliminatio