arXiv subjects
Publications and source records attributed to Florian Fürnsinn.
We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This includes a conjectural integrality of the sequence related to the Hörmander-Bernhardsson extremal function, for which we further give a direct proof as well.
We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Padé approximation. In conjunction with Honda's proof of the $p$-curvature conjecture for order one equations with polynomial coefficients we use this to deduce an effective version of the Grothendieck $p$-curvature conjecture for order one equations. More precisely, we bound the number of primes for which the $p$-curvature of a given differential equation has to vanish in terms of the height and the degree of the coefficients, in order to conclude it has a non-zero algebraic solution. Using this approach, we describe an algorithm that decides algebraicity of solutions of differential equation of order one using $p$-curvatures, and report on an implementation in SageMath.
We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p.
We report on implementations for algorithms treating algebraic and arithmetic properties of hypergeometric functions in the computer algebra system SageMath. We treat hypergeometric series over the rational numbers, over finite fields, and over the p-adics. Among other things, we provide implementations deciding algebraicity, computing valuations, and computing minimal polynomials in positive characteristic.
We compute the Galois groups of the reductions modulo the prime numbers $p$ of the generating series of Apéry numbers, Domb numbers and Almkvist--Zudilin numbers. We observe in particular that their behavior is governed by congruence conditions on p.
The aim of this paper is to investigate the algebraicity behavior of reductions of $D$-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric functions, it is known that their reductions modulo prime numbers, when defined, are algebraic. We formulate a conjecture that uniformizes the Galois groups of these reductions across different prime numbers. We then focus on hypergeometric functions, which serves as a test case for our conjecture. Refining the construction of an annihilating polynomial for the reduction of a hypergeometric function modulo a prime number p, we extract information on the respective Galois groups and show that they behave nicely as p varies.
We exhibit the joint symmetric distribution of the following two parameters on the set of unlabeled, simple, connected graphs with $n$ vertices. The first parameter is the maximal number of leaves attached to a vertex. The second parameter is the size of the largest set of vertices sharing the same closed neighborhood minus $1$. Apparently, this is the first example of a natural, non-trivial equidistribution of graph parameters on unlabeled connected graphs on a fixed set of vertices. Our proof is enumerative, using the theory of species. Exhibiting an explicit bijection interchanging the two parameters remains an open problem.
We provide a complete classification of the algebraicity of (generalized) hypergeometric functions with no restriction on the set of their parameters. Our characterization relies on the interlacing criteria of Christol (1987) and Beukers-Heckman (1989) for globally bounded and algebraic hypergeometric functions, however in a more general setting which allows arbitrary complex parameters with possibly integral differences. We also showcase the adapted criterion on a variety of different examples.
Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2, \ldots)(\!( x)\!)$ of $K(x)$, where $x'=1, z_1'=1/x$ and $z_i'=z_{i-1}'/z_{i-1}$. For differential equations of order $1$ it is shown that there exists a solution $y$ whose projections $y\vert_{z_{i+1}=z_{i+2}=\cdots=0}$ are algebraic over the field of rational functions $K(x, z_1, \ldots, z_{i})$ for all $i$. This can be seen as a characteristic $p$ analogue of Abel's problem about the algebraicity of logarithmic integrals. Further, the existence of infinite product representations of these solutions is shown. As a main tool $p^i$-curvatures are introduced, generalizing the notion of the $p$-curvature.
The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape of each solution. This completes partial investigations started by Taira Honda and Bernard Dwork. The main features are the introduction of a differential ring $\mathcal{R}$ in infinitely many variables mimicking the role of the (complex) iterated logarithms, and the proof that adding these "logarithms" already provides sufficiently many primitives so as to solve any differential equation with regular singularity in $\mathcal{R}$. A key step in the proof is the reduction of the involved differential operator to an Euler operator, its normal form, to solve Euler equations in $\mathcal{R}$ and to lift their (monomial) solutions to solutions of the original equation. The first (and already very striking) example of this outset is the exponential function $\exp_p$ in positive characteristic, solution of $y' = y$. We prove that it necessarily involves all variables and we construct its explicit (and quite mysterious) power series expansion. Additionally, relations of our results to the Grothendieck-Katz $p$-curvature conjecture and related conjectures will be discussed.