80 research outputs found
Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems
How many operations do we need on the average to compute an approximate root
of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked
whether a polynomial bound is possible, we prove a quasi-optimal bound
. This improves upon the previously known
bound.
The new algorithm relies on numerical continuation along \emph{rigid
continuation paths}. The central idea is to consider rigid motions of the
equations rather than line segments in the linear space of all polynomial
systems. This leads to a better average condition number and allows for bigger
steps. We show that on the average, we can compute one approximate root of a
random Gaussian polynomial system of~ equations of degree at most in
homogeneous variables with continuation steps. This is a
decisive improvement over previous bounds that prove no better than
continuation steps on the average
Computing periods of rational integrals
A period of a rational integral is the result of integrating, with respect to
one or several variables, a rational function over a closed path. This work
focuses particularly on periods depending on a parameter: in this case the
period under consideration satisfies a linear differential equation, the
Picard-Fuchs equation. I give a reduction algorithm that extends the
Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs
equations. The resulting algorithm is elementary and has been successfully
applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at
http://pierre.lairez.fr/supp/periods
A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time
We describe a deterministic algorithm that computes an approximate root of n
complex polynomial equations in n unknowns in average polynomial time with
respect to the size of the input, in the Blum-Shub-Smale model with square
root. It rests upon a derandomization of an algorithm of Beltr\'an and Pardo
and gives a deterministic affirmative answer to Smale's 17th problem. The main
idea is to make use of the randomness contained in the input itself
The boundary of the orbit of the 3 by 3 determinant polynomial
We consider the 3 by 3 determinant polynomial and we describe the limit
points of the set of all polynomials obtained from the determinant polynomial
by linear change of variables. This answers a question of J. M. Landsberg
Resolution except for minimal singularities II. The case of four variables
In this sequel to Resolution except for minimal singularities I, we find the
smallest class of singularities in four variables with which we necessarily end
up if we resolve singularities except for normal crossings. The main new
feature is a characterization of singularities in four variables which occur as
limits of triple normal crossings singularities, and which cannot be eliminated
by a birational morphism that avoids blowing up normal crossings singularities.Comment: 23 pages. Section 3 revised. Results unchange
Computing the homology of basic semialgebraic sets in weak exponential time
We describe and analyze an algorithm for computing the homology (Betti
numbers and torsion coefficients) of basic semialgebraic sets which works in
weak exponential time. That is, out of a set of exponentially small measure in
the space of data the cost of the algorithm is exponential in the size of the
data. All algorithms previously proposed for this problem have a complexity
which is doubly exponential (and this is so for almost all data)
Axioms for a theory of signature bases
Twenty years after the discovery of the F5 algorithm, Gr\"obner bases with
signatures are still challenging to understand and to adapt to different
settings. This contrasts with Buchberger's algorithm, which we can bend in many
directions keeping correctness and termination obvious. I propose an axiomatic
approach to Gr\"obner bases with signatures with the purpose of uncoupling the
theory and the algorithms, and giving general results applicable in many
different settings (e.g. Gr\"obner for submodules, F4-style reduction,
noncommutative rings, non-Noetherian settings, etc.)
Algorithms for minimal Picard-Fuchs operators of Feynman integrals
In even space-time dimensions the multi-loop Feynman integrals are integrals
of rational function in projective space. By using an algorithm that extends
the Griffiths--Dwork reduction for the case of projective hypersurfaces with
singularities, we derive Fuchsian linear differential equations, the
Picard--Fuchs equations, with respect to kinematic parameters for a large class
of massive multi-loop Feynman integrals. With this approach we obtain the
differential operator for Feynman integrals to high multiplicities and high
loop orders. Using recent factorisation algorithms we give the minimal order
differential operator in most of the cases studied in this paper. Amongst our
results are that the order of Picard--Fuchs operator for the generic massive
two-point -loop sunset integral in two-dimensions is
supporting the
conjecture that the sunset Feynman integrals are relative periods of
Calabi--Yau of dimensions . We have checked this explicitly till six
loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for
the massive five-point tardigrade non-planar two-loop integral in four
dimensions for generic mass and kinematic configurations, suggesting that it
arises from surface with Picard number 11. We determine as well
Picard--Fuchs operators of two-loop graphs with various multiplicities in four
dimensions, finding Fuchsian differential operators with either Liouvillian or
elliptic solutions.Comment: 57 pages. Results for differential operators are on the repository :
https://github.com/pierrevanhove/PicardFuchs#readm
Computing the Chow variety of quadratic space curves
Quadrics in the Grassmannian of lines in 3-space form a 19-dimensional
projective space. We study the subvariety of coisotropic hypersurfaces.
Following Gel'fand, Kapranov and Zelevinsky, it decomposes into Chow forms of
plane conics, Chow forms of pairs of lines, and Hurwitz forms of quadric
surfaces. We compute the ideals of these loci
Effective homology and periods of complex projective hypersurfaces
We provide an algorithm to compute an effective description of the homology
of complex projective hypersurfaces relying on Picard-Lefschetz theory. Next,
we use this description to compute high-precision numerical approximations of
the periods of the hypersurface. This is an improvement over existing
algorithms as this new method allows for the computation of periods of smooth
quartic surfaces in an hour on a laptop, which could not be attained with
previous methods. The general theory presented in this paper can be generalised
to varieties other than just hypersurfaces, such as elliptic fibrations as
showcased on an example coming from Feynman graphs. Our algorithm comes with a
SageMath implementation.Comment: 35 page
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