Alin Bostan
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View article: Tree-indexed sums of Catalan numbers
Tree-indexed sums of Catalan numbers Open
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 the…
View article: Algebraic solutions of linear differential equations: An arithmetic approach
Algebraic solutions of linear differential equations: An arithmetic approach Open
Given a linear differential equation with coefficients in , 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 m…
View article: A sharper multivariate Christol's theorem with applications to diagonals and Hadamard products
A sharper multivariate Christol's theorem with applications to diagonals and Hadamard products Open
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 independentl…
View article: Algebraic solutions of linear differential equations: an arithmetic approach
Algebraic solutions of linear differential equations: an arithmetic approach Open
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. Aft…
View article: On the $q$-Analogue of Pólya's Theorem
On the $q$-Analogue of Pólya's Theorem Open
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 a…
View article: Refined product formulas for Tamari intervals
Refined product formulas for Tamari intervals Open
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.
View article: Fast Algorithms for Discrete Differential Equations
Fast Algorithms for Discrete Differential Equations Open
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of so…
View article: Beating binary powering for polynomial matrices
Beating binary powering for polynomial matrices Open
The $N$th 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 …
View article: On the representability of sequences as constant terms
On the representability of sequences as constant terms Open
A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the…
View article: Minimization of differential equations and algebraic values of\n $E$-functions
Minimization of differential equations and algebraic values of\n $E$-functions Open
A power series being given as the solution of a linear differential equation\nwith appropriate initial conditions, minimization consists in finding a\nnon-trivial linear differential equation of minimal order having this power\nseries as a…
View article: Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems
Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems Open
We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gröbner bases are a classical…
View article: On some combinatorial sequences associated to invariant theory
On some combinatorial sequences associated to invariant theory Open
We study the enumerative and analytic properties of some sequences constructed using tensor invariant theory. The octant sequences are constructed from the exceptional Lie group $G_2$ and the quadrant sequences from the special linear grou…
View article: On an Integral Identity
On an Integral Identity Open
We give three elementary proofs of a nice equality of definite integrals, which arises from the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generali…
View article: Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems
Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems Open
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 dir…
View article: A Simple and Fast Algorithm for Computing the <i>N</i>-th Term of a Linearly Recurrent Sequence
A Simple and Fast Algorithm for Computing the <i>N</i>-th Term of a Linearly Recurrent Sequence Open
We present a simple and fast algorithm for computing the N-th term of a given linearly recurrent sequence. Our new algorithm uses O(M(d) log N) arithmetic operations, where d is the order of the recurrence, and M(d) denotes the number of a…
View article: Differential transcendence of Bell numbers and relatives: a Galois theoretic approach
Differential transcendence of Bell numbers and relatives: a Galois theoretic approach Open
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…
View article: A note on gamma triangles and local gamma vectors (with an appendix by Alin Bostan)
A note on gamma triangles and local gamma vectors (with an appendix by Alin Bostan) Open
This article introduces Gamma-triangles, which are closely related to F-triangles and H-triangles that were used in the combinatorial study of cluster complexes, and in some sense are more fundamental. We prove that Gamma-triangles can be …
View article: The generating function of Kreweras walks with interacting boundaries is not algebraic
The generating function of Kreweras walks with interacting boundaries is not algebraic Open
Beaton, Owczarek and Xu (2019) studied generating functions of Kreweras walks and of reverse Kreweras walks in the quarter plane, with interacting boundaries. They proved that for the reverse Kreweras step set, the generating function is a…
View article: A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence
A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence Open
We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(…
View article: Computing the N-th term of a q-holonomic sequence
Computing the N-th term of a q-holonomic sequence Open
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View article: Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series
Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series Open
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 Massazz…
View article: On sequences associated to the invariant theory of rank two simple Lie algebras
On sequences associated to the invariant theory of rank two simple Lie algebras Open
We study two families of sequences, listed in the On-Line Encyclopedia of Integer Sequences (OEIS), which are associated to invariant theory of Lie algebras. For the first family, we prove combinatorially that the sequences A059710 and A10…
View article: Subresultants of $(x-α)^m$ and $(x-β)^n$, Jacobi polynomials and complexity
Subresultants of $(x-α)^m$ and $(x-β)^n$, Jacobi polynomials and complexity Open
In an earlier article together with Carlos D'Andrea [BDKSV2017], we described explicit expressions for the coefficients of the order-$d$ polynomial subresultant of $(x-α)^m$ and $(x-β)^n $ with respect to Bernstein's set of polynomials $\{…