We provide short product formulas for the f-vectors of the canonical complexes of the Tamari lattices and of the cellular diagonals of the associahedra.

In 1984, Deligne proved that for any prime number p, the reduction modulo p of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in 𝔽_p. Moreover, he conjectured that the algebraic degrees d_p of these functions should grow at most polynomially in p. In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on d_p with an explicit and reasonable degree.

We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in 1/π with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane.

We provide a new arithmetic characterization for the sequence of coefficients of algebraic power series $f(t)$ having the property that the differential equation $y<^>{\prime}(t) = f(t) y(t)$ has algebraic solutions only. This extends a recent result by Delaygue and Rivoal, and also provides a new and shorter proof of an algebraicity result predicted by Golyshev.

The combination of recent results due to Yu and Chen [Proc. AMS 150(4), 2020, 1749-1765] and to Bostan and Yurkevich [Proc. AMS 150(5), 2022, 2131-2136] shows that the 3-D Euclidean shape of the square Clifford torus is uniquely determined by its isoperimetric ratio. This solves part of the still open uniqueness problem of the Canham model for biomembranes. In this work we investigate the generalization of the aforementioned result to the case of a rectangular Clifford torus. Like the square case, we find closed-form formulas in terms of hypergeometric functions for the isoperimetric ratio of its stereographic projection to ℝ^3 and show that the corresponding function is strictly increasing. But unlike the square case, we show that the isoperimetric ratio does not uniquely determine the Euclidean shape of a rectangular Clifford torus.

It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or transcendental, is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that is efficient in practice.

We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on to its complexity.

We consider singular (aka genus 0) walks in the quarter plane and their associated generating functions Q(x,y,t), which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables x and y) and of fixed length (encoded by the time variable t). We first prove that the previous series can be extended up to a universal value of t (in the sense that this holds for all singular models), namely t=1/2, and we provide a probabilistic interpretation of Q(x,y,1/2). As a second step, we refine earlier results in the literature and show that Q(x,y,t) is indeed differentially transcendental for any t∈(0,1/2]. Moreover, we prove that Q(x,y,1/2) is strongly differentially transcendental. As a last step, we show that for certain models the series expansion of Q(x,y,1/2) is directly related to Bernoulli numbers. This provides a second proof of its strong differential transcendence.

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments.

In 2003 Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $\mathbb{C}(\{t\})$ of meromorphic functions at $0$. We show that Klazar's result is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. We present the main principles of this theory in order to prove a general result about differential transcendence over $\mathbb{C}(\{t\})$, that we apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by~Klazar. Most of our examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak-Yeliussizov conjecture, according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.

Using an experimental mathematics approach, new relations are obtained between Dirichlet-like series for certain periodic coefficients and the moments of certain families of orthogonal polynomials.In addition to the classical hypergeometric orthogonal polynomials, of Racah type and continuous dual Hahn type, a new similar family of orthogonal polynomials intervenes.

We answer a question posed by Michael Aissen in 1979 about the $q$-analogue of a classical theorem of George Pólya (1922) on the algebraicity of (generalized) diagonals of bivariate rational power series. In particular, we prove that the answer to Aissen's question, in which he considers $q$ as a variable, is negative in general. Moreover, we show that the answer is positive if and only if $q$ is a root of unity.

The Nth power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in N. When Fast Fourier Transform (FFT) is available, the resulting complexity is softly linear in N, i.e. linear in N with extra logarithmic factors. We show that it is possible to beat binary powering, by an algorithm whose complexity is purely linear in N, even in absence of FFT. The key result making this improvement possible is that the entries of the Nth power of a polynomial matrix satisfy linear differential equations with polynomial coefficients whose orders and degrees are independent of N. Similar algorithms are proposed for two related problems: computing the Nth term of a C-finite sequence of polynomials, and modular exponentiation to the power N for bivariate polynomials.

A constant term sequence is a sequence of rational numbers whose n-th term is the constant term of Pn(x)Q(x), where P(x) and Q(x) are multivariate Laurent polynomials. While the generating functions of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify diagonals that are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We also consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms.

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial N! in arithmetic complexity quasi-linear in N. In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the N-th term of any holonomic sequence in essentially the same arithmetic complexity. We design q-analogues of these algorithms. We first extend Strassen's algorithm to the computation of the q-factorial of N, then Chudnovskys' algorithm to the computation of the N-th term of any q-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in N; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear q-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.

In this note, we propose a short and elementary proof of a non-vanishing result by Conca, Krattenthaler and Watanabe (2009).

We provide a new proof of the multivariate version of Christol's theorem about algebraic power series with coefficients in finite fields, as well as of its extension to perfect ground fields of positive characteristic obtained independently by Denef and Lipshitz, Sharif and Woodcok, and Harase. Our proof is elementary, effective, and allows for much sharper estimates. We discuss various applications of such estimates, in particular to a problem raised by Deligne concerning the algebraicity degree of reductions modulo $p$ of diagonals of multivariate algebraic power series with integer coefficients.

This work establishes exact formulae for the persistence probabilities pk(θ)=P[Y1⩾0,…,Yk⩾0] of an AR(1) sequence Yn=θYn−1+Xn, n=1,2,… with parameter θ∈R﹨(12,2) and symmetric uniform innovations Xn. The formulae are in terms of certain polynomials, most notably a family that arises in the case −1<θ<12 and was introduced by Mallows and Riordan in the very different context of counting finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results establish convolution-type factorizations in terms of the pk(θ) and their involutive conjugates pk(1/θ) for k=1,…,n and n⩾1. Regarding exact formulae for the pn(θ), these results are used for the cases θ<−1 and θ>2, but they are actually derived under more general conditions and therefore of independent interest, namely, one for AR(1) models with negative θ and continuous innovations, and a second one for AR(1) models with positive θ and continuous and symmetric innovations, the latter extending a classical universal formula of Sparre Andersen for symmetric random walks. We further explain why the case 12<θ<2 does not allow exact formulae for the pn(θ) as in the other cases and show that our results also lead to explicit asymptotic estimates for these probabilities.

Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimisation. Computing Grobner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Grobner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as Sparse-FGLM, designed by Faugere and Mou, is used to find a Grobner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is O(mD(2)), where D is the degree of the ideal generated by the input and m is the number of non-trivial columns of a certain D x D matrix. While asymptotic estimates are known for m in the case of generic polynomial systems, thus far, the complexity of Sparse-FGLM was unknown for the class of determinantal systems. By assuming Frob erg's conjecture, thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, we expand the work of Moreno-Socias by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity m by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the Sparse-FGLM algorithm for generic determinantal systems and, in particular, for generic critical point systems. We consider the ideal inside the polynomial ring K[x(1), ... , xn], where K is some infinite field, generated by p generic polynomials of degree d and the maximal minors of a p x (n - 1) polynomial matrix with generic entries of degree d - 1. Then, in this setting, for the case d = 2 and for n >> p we establish an exact formula for m in terms of n and p. Moreover, for d >= 3, we give a tight asymptotic formula, as n-+ infinity, for m in terms of n, p and d. (C) 2022 Elsevier Inc. All rights reserved.