arXiv subjects

Alin Bostan

Publications and source records attributed to Alin Bostan.

At least 37 records · Page 2Linked to original sources

arXiv2022

Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

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 tool for solving polynomial systems. For practical computations, this consists of two stages. First, a Gröbner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as \textsf{Sparse-FGLM}, designed by Faugère and Mou, is used to find a Gröbner basis of the same ideal but with respect to a lexicographic ordering. The complexity of this latter step, in terms of arithmetic operations, is $O(mD^2)$, where $D$ is the degree of the ideal and $m$ is the number of non-trivial columns of a certain $D \times D$ matrix. While asymptotic estimates are known for $m$ for generic polynomial systems, thus far, the complexity of \textsf{Sparse-FGLM} was unknown for determinantal systems. By assuming Fröberg's conjecture we expand the work of Moreno-Socías 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 \textsf{Sparse-FGLM} algorithm for generic determinantal systems and for generic critical point systems. We consider the ideal in the polynomial ring $\mathbb{K}[x_1, \dots, x_n]$, where $\mathbb{K}$ is some infinite field, generated by $p$ generic polynomials of degree $d$ and the maximal minors of a $p \times (n-1)$ polynomial matrix with generic entries of degree $d-1$. Then for the case $d=2$ and for $n \gg p$ we give an exact formula for $m$ in terms of $n$ and $p$. Moreover, for $d \geq 3$, we give an asymptotic formula, as $n \to \infty$, for $m$ in terms of $n,p$ and $d$.

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arXiv2021

A hypergeometric proof that ${\sf Iso}$ is bijective

We provide a short and elementary proof of the main technical result of the recent article "On the uniqueness of Clifford torus with prescribed isoperimetric ratio" by Thomas Yu and Jingmin Chen. The key of the new proof is an explicit expression of the central function (${\sf Iso}$, to be proved bijective) as a quotient of Gaussian hypergeometric functions.

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arXiv2021

The generating function of Kreweras walks with interacting boundaries is not algebraic

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 always algebraic, and for the Kreweras step set, the generating function is always D-finite. However, apart from the particular case where the interactions are symmetric in $x$ and~$y$, they left open the question of whether the latter one is algebraic. Using computer algebra tools, we confirm their intuition that the generating function of Kreweras walks is not algebraic, apart from the particular case already identified.

math.CO↗

arXiv2021

On a Class of Hypergeometric Diagonals

We prove that the diagonal of any finite product of algebraic functions of the form \begin{align*} {(1-x_1-\dots-x_n)^R}, \qquad R\in\mathbb{Q}, \end{align*} is a generalized hypergeometric function, and we provide an explicit description of its parameters. The particular case $(1-x-y)^R/(1-x-y-z)$ corresponds to the main identity of Abdelaziz, Koutschan and Maillard in [AKM2020, §3.2]. Our result is useful in both directions: on the one hand it shows that Christol's conjecture holds true for a large class of hypergeometric functions, on the other hand it allows for a very explicit and general viewpoint on the diagonals of algebraic functions of the type above. Finally, in contrast to [AKM2020], our proof is completely elementary and does not require any algorithmic help.

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arXiv2020

Improved algorithms for left factorial residues

We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that Kurepa's left factorial conjecture is still an open problem, as they show that there are no odd primes $p<2^{40}$ such that $p$ divides $!p$. Additionally, we confirm that there are no socialist primes $p$ with $5<p<2^{40}$.

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arXiv2020

Fast Computation of the $N$-th Term of a $q$-Holonomic Sequence and Applications

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. 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 $\sqrt{N}$; 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.

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arXiv2020

Stieltjes moment sequences for pattern-avoiding permutations

A small set of combinatorial sequences have coefficients that can be represented as moments of a nonnegative measure on $[0, \infty)$. Such sequences are known as Stieltjes moment sequences. This article focuses on some classical sequences in enumerative combinatorics, denoted $Av(\mathcal{P})$, and counting permutations of $\{1, 2, \ldots, n \}$ that avoid some given pattern $\mathcal{P}$. For increasing patterns $\mathcal{P}=(12\ldots k)$, we recall that the corresponding sequences, $Av(123\ldots k)$, are Stieltjes moment sequences, and we explicitly find the underlying density function, either exactly or numerically, by using the Stieltjes inversion formula as a fundamental tool. We show that the generating functions of the sequences $\, Av(1234)$ and $\, Av(12345)$ correspond, up to simple rational functions, to an order-one linear differential operator acting on a classical modular form given as a pullback of a Gaussian $\, _2F_1$ hypergeometric function, respectively to an order-two linear differential operator acting on the square of a classical modular form given as a pullback of a $\, _2F_1$ hypergeometric function. We demonstrate that the density function for the Stieltjes moment sequence $Av(123\ldots k)$ is closely, but non-trivially, related to the density attached to the distance traveled by a walk in the plane with $k-1$ unit steps in random directions. Finally, we study the challenging case of the $Av(1324)$ sequence and give compelling numerical evidence that this too is a Stieltjes moment sequence. Accepting this, we show how rigorous lower bounds on the growth constant of this sequence can be constructed, which are stronger than existing bounds. A further unproven assumption leads to even better bounds, which can be extrapolated to give an estimate of the (unknown) growth constant.

math.CO↗

arXiv2020

A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence

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 $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation, powering of matrices and high-order lifting.

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arXiv2020

Explicit degree bounds for right factors of linear differential operators

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, we get degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.

math.CA↗

arXiv2020

On an integral identity

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 generalization together with an equally elementary proof and discuss some consequences.

math.CA↗

arXiv2019

On sequences associated to the invariant theory of rank two simple Lie algebras

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 A108307 are related by a binomial transform. Based on this, we present two independent proofs of a recurrence equation for A059710, which was conjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs' conjecture by the method of algebraic residues. As a consequence, closed formulae for the generating function of sequence A059710 are obtained in terms of classical Gaussian hypergeometric functions. Moreover, we show that sequences in the second family are also related by binomial transforms.

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arXiv2019

On the least common multiple of several random integers

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ for a wide class of multiplicative arithmetic functions~$f$ with polynomial growth $r>-1$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).

math.PR↗

arXiv2018

Fast Coefficient Computation for Algebraic Power Series in Positive Characteristic

We revisit Christol's theorem on algebraic power series in positive characteristic and propose yet another proof for it. This new proof combines several ingredients and advantages of existing proofs, which make it very well-suited for algorithmic purposes. We apply the construction used in the new proof to the design of a new efficient algorithm for computing the $N$th coefficient of a given algebraic power series over a perfect field of characteristic~$p$. It has several nice features: it is more general, more natural and more efficient than previous algorithms. Not only the arithmetic complexity of the new algorithm is linear in $\log N$ and quasi-linear in~$p$, but its dependency with respect to the degree of the input is much smaller than in the previously best algorithm. {Moreover, when the ground field is finite, the new approach yields an even faster algorithm, whose bit complexity is linear in $\log N$ and quasi-linear in~$\sqrt{p}$}.

math.NT↗

arXiv2018

Counting walks with large steps in an orthant

In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions ofquadrant walks are now well-understood, provided the allowed steps aresmall, that is $S \subset \{-1, 0,1\}^2$. In particular, having smallsteps is crucial for the definition of a certain group of bi-rationaltransformations of the plane. It has been proved that this group is finite ifand only if the corresponding generating function is D-finite (that is, it satisfies a lineardifferential equation with polynomial coefficients). This group is also thekey to the uniform solution of 19 of the 23 small step models possessing afinite group.In contrast, almost nothing is known for walks with arbitrary steps. In thispaper, we extend the definition of the group, or rather of the associatedorbit, to this general case, and generalize the above uniform solution ofsmall step models. When this approach works, it invariably yields a D-finitegenerating function. We apply it to many quadrant problems, including some infinite families.After developing the general theory, we consider the $13\ 110$ two-dimensionalmodels with steps in $\{-2,-1,0,1\}^2$ having at least one $-2$ coordinate. Weprove that only 240 of them have a finite orbit, and solve 231 of them withour method. The 9 remaining models are the counterparts of the 4 models of thesmall step case that resist the uniform solution method (and which are knownto have an algebraic generating function). We conjecture D-finiteness for their generatingfunctions, but only two of them are likely to be algebraic. We also provenon-D-finiteness for the $12\ 870$ models with an infinite orbit, except for16 of them.

math.CO↗

arXiv2018

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.

cs.SC↗

arXiv2017

Subresultants in multiple roots: an extremal case

We provide explicit formulae for the coefficients of the order-d polynomial subresultant of (x-α)^m and (x-β)^n with respect to the set of Bernstein polynomials \{(x-α)^j(x-β)^{d-j}, \, 0\le j\le d\}. They are given by hypergeometric expressions arising from determinants of binomial Hankel matrices.

math.AC↗

arXiv2017

On matrices with displacement structure: generalized operators and faster algorithms

For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product $\mathsf{A}\mathsf{B}$, where $\mathsf{A}$ is a structured $n \times n$ matrix of displacement rank $α$, and $\mathsf{B}$ is an arbitrary $n\timesα$ matrix. Given $\mathsf{B}$ and a so-called "generator" of $\mathsf{A}$, this product is classically computed with a cost ranging from $O(α^2 \mathscr{M}(n))$ to $O(α^2 \mathscr{M}(n)\log(n))$ arithmetic operations, depending on the type of structure of $\mathsf{A}$; here, $\mathscr{M}$ is a cost function for polynomial multiplication. In this paper, we first generalize classical displacement operators, based on block diagonal matrices with companion diagonal blocks, and then design fast algorithms to perform the task above for this extended class of structured matrices. The cost of these algorithms ranges from $O(α^{ω-1} \mathscr{M}(n))$ to $O(α^{ω-1} \mathscr{M}(n)\log(n))$, with $ω$ such that two $n \times n$ matrices over a field can be multiplied using $O(n^ω)$ field operations. By combining this result with classical randomized regularization techniques, we obtain faster Las Vegas algorithms for structured inversion and linear system solving.

cs.SC↗

arXiv2016

Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or $-1$. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane $\mathbb{N}^2$, counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201--215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their $19 \times 4$ combinatorially meaningful specializations only four are algebraic functions.

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