8,836 research outputs found
A simple approach to counterterms in N=8 supergravity
We present a simple systematic method to study candidate counterterms in N=8
supergravity. Complicated details of the counterterm operators are avoided
because we work with the on-shell matrix elements they produce. All n-point
matrix elements of an independent SUSY invariant operator of the form D^{2k}
R^n +... must be local and satisfy SUSY Ward identities. These are strong
constraints, and we test directly whether or not matrix elements with these
properties can be constructed. If not, then the operator does not have a
supersymmetrization, and it is excluded as a potential counterterm. For n>4, we
find that R^n, D^2 R^n, D^4 R^n, and D^6 R^n are excluded as counterterms of
MHV amplitudes, while only R^n and D^2 R^n are excluded at the NMHV level. As a
consequence, for loop order L<7, there are no independent D^{2k}R^n
counterterms with n>4. If an operator is not ruled out, our method constructs
an explicit superamplitude for its matrix elements. This is done for the 7-loop
D^4 R^6 operator at the NMHV level and in other cases. We also initiate the
study of counterterms without leading pure-graviton matrix elements, which can
occur beyond the MHV level. The landscape of excluded/allowed candidate
counterterms is summarized in a colorful chart.Comment: 25 pages, 1 figure, published versio
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
Counting Value Sets: Algorithm and Complexity
Let be a prime. Given a polynomial in \F_{p^m}[x] of degree over
the finite field \F_{p^m}, one can view it as a map from \F_{p^m} to
\F_{p^m}, and examine the image of this map, also known as the value set. In
this paper, we present the first non-trivial algorithm and the first complexity
result on computing the cardinality of this value set. We show an elementary
connection between this cardinality and the number of points on a family of
varieties in affine space. We then apply Lauder and Wan's -adic
point-counting algorithm to count these points, resulting in a non-trivial
algorithm for calculating the cardinality of the value set. The running time of
our algorithm is . In particular, this is a polynomial time
algorithm for fixed if is reasonably small. We also show that the
problem is #P-hard when the polynomial is given in a sparse representation,
, and is allowed to vary, or when the polynomial is given as a
straight-line program, and is allowed to vary. Additionally, we prove
that it is NP-hard to decide whether a polynomial represented by a
straight-line program has a root in a prime-order finite field, thus resolving
an open problem proposed by Kaltofen and Koiran in
\cite{Kaltofen03,KaltofenKo05}
Effective Invariant Theory of Permutation Groups using Representation Theory
Using the theory of representations of the symmetric group, we propose an
algorithm to compute the invariant ring of a permutation group. Our approach
have the goal to reduce the amount of linear algebra computations and exploit a
thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at
http://www.springer.com
A local construction of the Smith normal form of a matrix polynomial
We present an algorithm for computing a Smith form with multipliers of a
regular matrix polynomial over a field. This algorithm differs from previous
ones in that it computes a local Smith form for each irreducible factor in the
determinant separately and then combines them into a global Smith form, whereas
other algorithms apply a sequence of unimodular row and column operations to
the original matrix. The performance of the algorithm in exact arithmetic is
reported for several test cases.Comment: 26 pages, 6 figures; introduction expanded, 10 references added, two
additional tests performe
Sparse multivariate polynomial interpolation in the basis of Schubert polynomials
Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in
the study of cohomology rings of flag manifolds in 1980's. These polynomials
generalize Schur polynomials, and form a linear basis of multivariate
polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials,
which generalize skew Schur polynomials, and expand in the Schubert basis with
the generalized Littlewood-Richardson coefficients.
In this paper we initiate the study of these two families of polynomials from
the perspective of computational complexity theory. We first observe that skew
Schubert polynomials, and therefore Schubert polynomials, are in \CountP
(when evaluating on non-negative integral inputs) and \VNP.
Our main result is a deterministic algorithm that computes the expansion of a
polynomial of degree in in the basis of Schubert
polynomials, assuming an oracle computing Schubert polynomials. This algorithm
runs in time polynomial in , , and the bit size of the expansion. This
generalizes, and derandomizes, the sparse interpolation algorithm of symmetric
polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied
Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general
enough to accommodate any linear basis satisfying certain natural properties.
Applications of the above results include a new algorithm that computes the
generalized Littlewood-Richardson coefficients.Comment: 20 pages; some typos correcte
Interval structure of the Pieri formula for Grothendieck polynomials
We give a combinatorial interpretation of a Pieri formula for double
Grothendieck polynomials in terms of an interval of the Bruhat order. Another
description had been given by Lenart and Postnikov in terms of chain
enumerations. We use Lascoux's interpretation of a product of Grothendieck
polynomials as a product of two kinds of generators of the 0-Hecke algebra, or
sorting operators. In this way we obtain a direct proof of the result of Lenart
and Postnikov and then prove that the set of permutations occuring in the
result is actually an interval of the Bruhat order.Comment: 27 page
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