Value Distribution of Elementary Symmetric Polynomials and Their Perturbations Over Finite Fields

مرکز و کتابخانه مطالعات اسلامی به زبان های اروپایی

منو

" Value Distribution of Elementary Symmetric Polynomials and Their Perturbations Over Finite Fields "
Rapello, César A. Serna Medina, Luis A.

Document Type	:	Latin Dissertation
Language of Document	:	English
Record Number	:	1107072
Doc. No	:	TLpq2428656740
Main Entry	:	Medina, Luis A.
	:	Rapello, César A. Serna
Title & Author	:	Value Distribution of Elementary Symmetric Polynomials and Their Perturbations Over Finite Fields\ Rapello, César A. SernaMedina, Luis A.
College	:	University of Puerto Rico, Rio Piedras (Puerto Rico)
Date	:	2020
student score	:	2020
Degree	:	Ph.D.
Page No	:	62
Abstract	:	In this work we present two studies related to exponential sums and Walsh transformations of q-ary functions. In our first study we establish the asymptotic behavior of some generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the elementary symmetric polynomial of degree k (and its perturbations) returns β∈ Fq where Fq represents the field of q elements. Our study extends many of the results known for perturbations over the binary field to any finite field. In particular, we establish when a particular perturbation is asymptotically balanced over a prime field and provide a construction to find such perturbations over any finite field. Our second study is related to generalized p-ary functions. In particular, we show that Walsh-Hadamard transformations of generalized p-ary functions whose components are symmetric, rotation symmetric or a combination or concatenation of them are C-finite sequences. This result generalizes many of the known results for regular p-ary functions. We also present a study of the roots of the characteristic polynomials related to these sequences and show that properties like balancedness and being bent are not shared by the underline p-ary functions.
Subject	:	Applied mathematics
	:	Mathematics