16 citations · 59 across the 20 of their papers we have counts for
23 papers
The theory of maximal Hardy fields
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We show that all maximal Hardy fields are elementarily equivalent as differential fields to the differential field of transseries, and give various applications of this…
Constructing -free Hardy fields
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We show that every Hardy field extends to an -free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations.…
Normalizing Asymptotic Differential Equations
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection wit…
Fast interpolation of sparse multivariate polynomials
Joris van der Hoeven, Grégoire Lecerf
Consider a sparse multivariate polynomial f with integer coefficients. Assume that f is represented as a "modular black box polynomial", e.g. via an algorithm to evaluate f at arbi…
Factoring sparse polynomials fast
Alexander Demin, Joris van der Hoeven
Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms…
Surreal numbers as hyperseries
Vincent Bagayoko, Joris van der Hoeven
Surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all ordinal numbers. Hyperseries can be r…
The hyperserial field of surreal numbers
Vincent Bagayoko, Joris van der Hoeven
For any ordinal , we show how to define a hyperexponential and a hyperlogarithm on the class of positive infinitely large surreal…
Filling gaps in Hardy fields
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We show how to fill "countable" gaps in Hardy fields. We use this to prove that any two maximal Hardy fields are back-and-forth equivalent.
Surreal substructures
Vincent Bagayoko, Joris van der Hoeven
Conway's field No of surreal numbers comes both with a natural total order and an additional "simplicity relation" which is also a partial order. Considering No as a doubly ordered…
Maximal Hardy Fields
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We show that all maximal Hardy fields are elementarily equivalent as differential fields, and give various applications of this result and its proof. We also answer some questions…
Optimizing the half-gcd algorithm
Joris van der Hoeven
In this paper, we propose a carefully optimized "half-gcd" algorithm for polynomials. We achieve a constant speed-up with respect to previous work for the asymptotic time complexit…
Revisiting closed asymptotic couples
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
Every discrete definable subset of a closed asymptotic couple with ordered scalar field is shown to be contained in a finite-dimensional -linear subs…
Maximal immediate extensions of valued differential fields
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We show that every valued differential field has an immediate strict extension that is spherically complete. We also discuss the issue of uniqueness up to isomorphism of such an ex…
Faster integer multiplication using plain vanilla FFT primes
David Harvey, Joris van der Hoeven
Assuming a conjectural upper bound for the least prime in an arithmetic progression, we show that n-bit integers may be multiplied in O(n log n 4^(log^* n)) bit operations.
Dimension in the realm of transseries
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
Let be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of , also in relation to its codi…
The surreal numbers as a universal H-field
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We show that the natural embedding of the differential field of transseries into Conway's field of surreal numbers with the Berarducci-Mantova derivation is an elementary embedding…
Asymptotic Differential Algebra and Model Theory of Transseries
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
We develop here the algebra of the differential field of transseries and of related valued differential fields. This book contains in particular our recently obtained decisive posi…
Modular SIMD arithmetic in Mathemagix
Joris van der Hoeven, Grégoire Lecerf, Guillaume Quintin
Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of…
Faster polynomial multiplication over finite fields
David Harvey, Joris van der Hoeven, Grégoire Lecerf
Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n. For n large compared to p, we establish the bound M_p(n)…
Even faster integer multiplication
David Harvey, Joris van der Hoeven, Grégoire Lecerf
We give a new proof of Fürer's bound for the cost of multiplying n-bit integers in the bit complexity model. Unlike Fürer, our method does not require constructing special coeffici…
Quasi-optimal multiplication of linear differential operators
Alexandre Benoit, Alin Bostan, Joris van der Hoeven
We show that linear differential operators with polynomial coefficients over a field of characteristic zero can be multiplied in quasi-optimal time. This answers an open question r…
Towards a Model Theory for Transseries
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures,…
Automatic asymptotics for coefficients of smooth, bivariate rational functions
Timothy DeVries, Joris van der Hoeven, Robin Pemantle
We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q…