Extended Rate, more GFUN

We present a software package that guesses formulae for sequences of, for example, rational numbers or rational functions, given the first few terms. We implement an algorithm due to Bernhard Beckermann and George Labahn, together with some enhancements to render our package efficient. Thus we extend and complement Christian Krattenthaler's program Rate, the parts concerned with guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's qGeneratingFunctions.m.Comment: 26 page

The computational challenge of enumerating high-dimensional rook walks

We provide guessed recurrence equations for the counting sequences of rook paths on d-dimensional chess boards starting at (0..0) and ending at (n..n), where d=2,3,...,12. Our recurrences suggest refined asymptotic formulas of these sequences. Rigorous proofs of the guessed recurrences as well as the suggested asymptotic forms are posed as challenges to the reader

Hankel Determinant Calculus for the Thue-Morse and related sequences

The Hankel determinants of certain automatic sequences $f$ are evaluated, based on a calculation modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as $LU$-decompo\-si\-tion and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence $g$ equal to $f$ modulo $p$ are equal to the Hankel determinants of $f$ modulo $p$. The clue then consists of finding a nice sequence $g$, whose Hankel determinants have closed-form expressions. Several examples are presented, including a result saying that the Hankel determinants of the Thue-Morse sequence are nonzero, first proved by Allouche, Peyri\`ere, Wen and Wen using determinant manipulation. The present approach shortens the proof of the latter result significantly. We also prove that the corresponding Hankel determinants do not vanish when the powers $2^n$ in the infinite product defining the $\pm 1$ Thue--Morse sequence are replaced by $3^n$

Operations for D-algebraic Functions

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass elliptic functions, and holonomic or D-finite functions are D-algebraic. These functions form a field, and are closed under composition, taking functional inverse, and derivation. We present implementation for each underlying operation. We also give a systematic way for computing an algebraic differential equation from a linear differential equation with D-finite function coefficients. Each command is a feature of our Maple package $NLDE$ available at https://mathrepo.mis.mpg.de/OperationsForDAlgebraicFunctions.Comment: 4 pages + 1 (10 references). Mathematical software pape

The noncommutative ward metric

We analyze the moduli-space metric in the static non-Abelian charge-two sector of the Moyal-deformed CP1 sigma model in 1 + 2 dimensions. After carefully reviewing the commutative results of Ward and Ruback, the noncommutative Kähler potential is expanded in powers of dimensionless moduli. In two special cases we sum the perturbative series to analytic expressions. For any nonzero value of the noncommutativity parameter, the logarithmic singularity of the commutative metric is expelled from the origin of the moduli space and possibly altogether.DAA

Combinatorial methods in differential algebra

This thesis studies various aspects of differential algebra, from fundamental concepts to practical computations. A characteristic feature of this work is the use of combinatorial techniques, which offer a unique and original perspective on the subject matter. First, we establish the connection between the n-jet space of the fat point defined by xm and the stable set polytope of a perfect graph. We prove that the dimension of the coordinate ring of the scheme defined by polynomial arcs of degree less than or equal to n is a polynomial in m of degree n + 1. This is based on Zobnin’s result which states that the set {x^m} is a differential Gr ̈obner basis for its differential ideal. We generalize this statement to the case of two independent variables and link the dimensions in this case to some triangulations of the p × q rectangle, where the pair (p, q) now plays the role of n. Second, we study the arc space of the fat point x^m on a line from the point of view of filtration by finite-dimensional differential algebras. We prove that the generating series of the dimensions of these differential algebras is m/(1 -mt) . Based on this we propose a definition of the multiplicity of a solution of an algebraic differential equation as the growth of the dimensions of these differential algebras. This generalizes the concept of the multiplicity of an ideal in a polynomial ring. Furthermore, we determine a full description of the set of standard monomials of the differential ideal generated by x^m. This description proves a conjecture by Afsharijoo concerning a new version of the Roger-Ramanujan identities. Every homogeneous linear system of partial differential equations with constant coef- ficients can be encoded by a submodule of the ring of polynomials. We develop practical methods for computing the space of solutions to these PDEs. These spaces are typically infinite dimensional, and we use the Ehrenpreis–Palamodov Theorem for finite encoding. We apply this finite encoding to the solutions of the PDEs associated with the arc spaces of a double point. We prove that these vector spaces are spanned by determinants of some special Wronskians, and we relate them to differentially homogeneous polynomials. Finally, we introduce D-algebraic functions: they are solutions to algebraic differential equations. We study closure properties of these functions. We present practical algorithms and their implementations for carrying out arithmetic operations on D-algebraic functions. This amounts to solving elimination problems for differential ideals