153 research outputs found
Reducing the Number of Homogeneous Linear Equations in Finding Annihilators
Given a Boolean function on -variables, we find a reduced set of homogeneous linear equations by solving which one can decide whether there exist annihilators at degree or not.
Using our method the size of the associated matrix becomes
, where,
and
and the time required to construct the matrix is same as the size of the matrix. This is a
preprocessing step before the exact solution strategy (to decide on the existence of the annihilators) that requires to solve the set of homogeneous linear equations (basically to calculate the rank) and this can be improved when the number of variables and the number of equations are minimized. As the linear transformation on the input variables of the Boolean function keeps the degree of the annihilators invariant, our preprocessing step can be more efficiently applied if one can find an affine transformation over to get such that is maximized (and in turn is minimized too). We present an efficient heuristic towards this. Our study also shows for what kind of Boolean functions the asymptotic reduction in the size of the matrix is possible and when the reduction is not asymptotic but constant
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity
Testing whether a set of polynomials has an algebraic dependence
is a basic problem with several applications. The polynomials are given as
algebraic circuits. Algebraic independence testing question is wide open over
finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is
NP (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we
put the problem in AM coAM. In particular, dependence testing is
unlikely to be NP-hard and joins the league of problems of "intermediate"
complexity, eg. graph isomorphism & integer factoring. Our proof method is
algebro-geometric-- estimating the size of the image/preimage of the polynomial
map over the finite field. A gap in this size is utilized in the
AM protocols.
Next, we study the open question of testing whether every annihilator of
has zero constant term (Kayal, CCC'09). We give a geometric
characterization using Zariski closure of the image of ;
introducing a new problem called approximate polynomials satisfiability (APS).
We show that APS is NP-hard and, using projective algebraic-geometry ideas, we
put APS in PSPACE (prior best was EXPSPACE via Grobner basis computation). As
an unexpected application of this to approximative complexity theory we get--
Over any field, hitting-set for can be designed in PSPACE.
This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly
mitigating the GCT Chasm (exponentially in terms of space complexity)
A Geometric Algorithm for the Factorization of Spinor Polynomials
We present a new algorithm to decompose generic spinor polynomials into
linear factors. Spinor polynomials are certain polynomials with coefficients in
the geometric algebra of dimension three that parametrize rational conformal
motions. The factorization algorithm is based on the "kinematics at infinity"
of the underlying rational motion. Factorizations exist generically but not
generally and are typically not unique. We prove that generic multiples of
non-factorizable spinor polynomials admit factorizations and we demonstrate at
hand of an example how our ideas can be used to tackle the hitherto unsolved
problem of "factorizing" algebraic motions
Feynman Integrals and Intersection Theory
We introduce the tools of intersection theory to the study of Feynman
integrals, which allows for a new way of projecting integrals onto a basis. In
order to illustrate this technique, we consider the Baikov representation of
maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of
differential forms with logarithmic singularities on the boundaries of the
corresponding integration cycles. We give an algorithm for computing a basis
decomposition of an arbitrary maximal cut using so-called intersection numbers
and describe two alternative ways of computing them. Furthermore, we show how
to obtain Pfaffian systems of differential equations for the basis integrals
using the same technique. All the steps are illustrated on the example of a
two-loop non-planar triangle diagram with a massive loop.Comment: 13 pages, published versio
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
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