4,100 research outputs found
On the Structure of Quintic Polynomials
We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let F=F_q be a prime field. Suppose f:F^n to F is a degree five polynomial with bias(f)=delta. We prove the following two structural properties for such f.
1. We have f= sum_{i=1}^{c} G_i H_i + Q, where G_i and H_is are nonconstant polynomials satisfying deg(G_i)+deg(H_i)<= 5 and Q is a degree <5 polynomial. Moreover, c does not depend on n.
2. There exists an Omega_{delta,q}(n) dimensional affine subspace V subseteq F^n such that f|_V is a constant.
Cohen and Tal [Random 2015] proved that biased polynomials of degree at most four are constant on a subspace of dimension Omega(n). Item 2.]extends this to degree five polynomials. A corollary to Item 2. is that any degree five affine disperser for dimension k is also an affine extractor for dimension O(k). We note that Item 2. cannot hold for degrees six or higher.
We obtain our results for degree five polynomials as a special case of structure theorems that we prove for biased degree d polynomials when d<|F|+4. While the d<|F|+4 assumption seems very restrictive, we note that prior to our work such structure theorems were only known for d<|F| by Green and Tao [Contrib. Discrete Math. 2009] and Bhowmick and Lovett [arXiv:1506.02047]. Using algorithmic regularity lemmas for polynomials developed by Bhattacharyya, et al. [SODA 2015], we show that whenever such a strong structure exists, it can be found algorithmically in time polynomial in n
Topological String Partition Functions as Polynomials
We investigate the structure of the higher genus topological string
amplitudes on the quintic hypersurface. It is shown that the partition
functions of the higher genus than one can be expressed as polynomials of five
generators. We also compute the explicit polynomial forms of the partition
functions for genus 2, 3, and 4. Moreover, some coefficients are written down
for all genus.Comment: 22 pages, 6 figures. v2:typos correcte
Three Generations on the Quintic Quotient
A three-generation SU(5) GUT, that is 3x(10+5bar) and a single 5-5bar pair,
is constructed by compactification of the E_8 heterotic string. The base
manifold is the Z_5 x Z_5-quotient of the quintic, and the vector bundle is the
quotient of a positive monad. The group action on the monad and its
bundle-valued cohomology is discussed in detail, including topological
restrictions on the existence of equivariant structures. This model and a
single Z_5 quotient are the complete list of three generation quotients of
positive monads on the quintic.Comment: 19 pages, LaTeX. v2: section on anomaly cancellation adde
On Klein's Icosahedral Solution of the Quintic
We present an exposition of the icosahedral solution of the quintic equation
first described in Klein's classic work "Lectures on the icosahedron and the
solution of equations of the fifth degree". Although we are heavily influenced
by Klein we follow a slightly different approach which enables us to arrive at
the solution more directly.Comment: 29 pages, 5 figure
R Symmetries in the Landscape
In the landscape, states with symmetries at the classical level form a
distinct branch, with a potentially interesting phenomenology. Some preliminary
analyses suggested that the population of these states would be significantly
suppressed. We survey orientifolds of IIB theories compactified on Calabi-Yau
spaces based on vanishing polynomials in weighted projective spaces, and find
that the suppression is quite substantial. On the other hand, we find that a
R-parity is a common feature in the landscape. We discuss whether the
cosmological constant and proton decay or cosmology might select the low energy
branch. We include also some remarks on split supersymmetry.Comment: 13 page
Deforming, revolving and resolving - New paths in the string theory landscape
In this paper we investigate the properties of series of vacua in the string
theory landscape. In particular, we study minima to the flux potential in type
IIB compactifications on the mirror quintic. Using geometric transitions, we
embed its one dimensional complex structure moduli space in that of another
Calabi-Yau with h^{1,1}=86 and h^{2,1}=2. We then show how to construct
infinite series of continuously connected minima to the mirror quintic
potential by moving into this larger moduli space, applying its monodromies,
and moving back. We provide an example of such series, and discuss their
implications for the string theory landscape.Comment: 41 pages, 5 figures; minor corrections, published versio
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