2,740 research outputs found
On the Symmetries of and Equivalence Test for Design Polynomials
In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [Neeraj Kayal et al., 2014], is the following: NW_{d,k}({x}) = sum_{h in F_d[z], deg(h) <= k}{ prod_{i=0}^{d-1}{x_{i, h(i)}}}, where d is a prime, F_d is the finite field with d elements, and k << d. The degree of the gcd of every pair of monomials in NW_{d,k} is at most k. For concreteness, we fix k = ceil[sqrt{d}]. The family of polynomials NW := {NW_{d,k} : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of NW beyond the fact that NW in VNP. Is NW_{d,k} characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW_{d,k}? What is the complexity of equivalence test for NW, i.e., given black-box access to a f in F[{x}], can we check efficiently if there exists an invertible linear transformation A such that f = NW_{d,k}(A * {x})? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third.
We show that NW_{d,k} is characterized by its group of symmetries over C, but not over R. We also show that NW_{d,k} is characterized by circuit identities which implies that NW_{d,k} is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for NW.
We give an efficient equivalence test for NW in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW_{d,k}: We show that if A is in the group of symmetries of NW_{d,k} then A = D * P, where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW_{d,k}, and using an interplay between the Hessian of NW_{d,k} and the evaluation dimension
Complexity spectrum of some discrete dynamical systems
We first study birational mappings generated by the composition of the matrix
inversion and of a permutation of the entries of matrices. We
introduce a semi-numerical analysis which enables to compute the Arnold
complexities for all the possible birational transformations. These
complexities correspond to a spectrum of eighteen algebraic values. We then
drastically generalize these results, replacing permutations of the entries by
homogeneous polynomial transformations of the entries possibly depending on
many parameters. Again it is shown that the associated birational, or even
rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil
A framework for structured linearizations of matrix polynomials in various bases
We present a framework for the construction of linearizations for scalar and
matrix polynomials based on dual bases which, in the case of orthogonal
polynomials, can be described by the associated recurrence relations. The
framework provides an extension of the classical linearization theory for
polynomials expressed in non-monomial bases and allows to represent polynomials
expressed in product families, that is as a linear combination of elements of
the form , where and
can either be polynomial bases or polynomial families
which satisfy some mild assumptions. We show that this general construction can
be used for many different purposes. Among them, we show how to linearize sums
of polynomials and rational functions expressed in different bases. As an
example, this allows to look for intersections of functions interpolated on
different nodes without converting them to the same basis. We then provide some
constructions for structured linearizations for -even and
-palindromic matrix polynomials. The extensions of these constructions
to -odd and -antipalindromic of odd degree is discussed and
follows immediately from the previous results
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
Magic composite pulses
I describe composite pulses during which the average dipolar interactions
within a spin ensemble are controlled while realizing a global rotation. The
construction method used is based on the average Hamiltonian theory and rely on
the geometrical properties of the spin-spin dipolar interaction only. I present
several such composite pulses robust against standard experimental defects in
NRM: static or radio-frequency field miscalibration, fields inhomogeneities.
Numerical simulations show that the magic sandwich pulse sequence, a pulse
sequence that reverse the average dipolar field while applied, is plagued by
defects originating from its short initial and final \pi/2 radio-frequency
pulses. Using the magic composite pulses instead of \pi/2 pulses improves the
magic sandwich effect. A numerical test using a classical description of NMR
allows to check the validity of the magic composite pulses and estimate their
efficiency.Comment: 22 pages, 6 figure
Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method
Given the matrix polynomial , we
consider the associated polynomial eigenvalue problem. This problem, viewed in
terms of computing the roots of the scalar polynomial , is treated
in polynomial form rather than in matrix form by means of the Ehrlich-Aberth
iteration. The main computational issues are discussed, namely, the choice of
the starting approximations needed to start the Ehrlich-Aberth iteration, the
computation of the Newton correction, the halting criterion, and the treatment
of eigenvalues at infinity. We arrive at an effective implementation which
provides more accurate approximations to the eigenvalues with respect to the
methods based on the QZ algorithm. The case of polynomials having special
structures, like palindromic, Hamiltonian, symplectic, etc., where the
eigenvalues have special symmetries in the complex plane, is considered. A
general way to adapt the Ehrlich-Aberth iteration to structured matrix
polynomial is introduced. Numerical experiments which confirm the effectiveness
of this approach are reported.Comment: Submitted to Linear Algebra App
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