scholarly journals A power sum formula by Carlitz and its applications to permutation rational functions of finite fields

2021 ◽  
Author(s):  
Xiang-dong Hou
Keyword(s):  
Finite Fields ◽  
Rational Functions ◽  
Power Sum ◽  
Sum Formula

scholarly journals ELEMENTS OF HIGH ORDER ON FINITE FIELDS FROM ELLIPTIC CURVES

2010 ◽  
Vol 81 (3) ◽  
pp. 425-429 ◽  
Author(s):  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Rational Functions ◽  
Large Degree ◽  
High Order ◽  
Prime Field ◽  
Nontrivial Estimate ◽  
Special Cases

AbstractWe discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

scholarly journals The Power-Sum Formula and the Bernoullian Function

1912 ◽  
Vol 6 (99) ◽  
pp. 332 ◽  
Author(s):  
W. F. Sheppard
Keyword(s):  
Power Sum ◽  
Sum Formula

scholarly journals Nets obtained from rational functions over finite fields

1993 ◽  
Vol 63 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Gerhard Larcher
Keyword(s):  
Finite Fields ◽  
Rational Functions

scholarly journals Subgroups generated by rational functions in finite fields

2014 ◽  
Vol 176 (2) ◽  
pp. 241-253 ◽  
Author(s):  
Domingo Gómez-Pérez ◽  
Igor E. Shparlinski
Keyword(s):  
Finite Fields ◽  
Rational Functions

MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS

2005 ◽  
Vol 01 (01) ◽  
pp. 1-32 ◽  
Author(s):  
J. BOURGAIN
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Rational Functions ◽  
Exponential Sum ◽  
Kloosterman Sums ◽  
Product Theorem ◽  
Short Intervals ◽  
Prime Fields ◽  
Product Sets ◽  
Mod P

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if [Formula: see text] and pε < |B|, |C| < |A| < p1-ε, then |A + B| + |A · C| > pδ (ε)|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from [Formula: see text] to [Formula: see text] satisfies |F(A,B)| > pβ for some β = β (α) > α whenever [Formula: see text] and |A| ~ |B|~ pα, 0 < α < 1. The exponential sum ∑x∈ A,y∈Bεp(axy+bx2y2), ab ≠ 0 ( mod p), may be estimated nontrivially for arbitrary sets [Formula: see text] satisfying |A|, |B| > pρ where ρ < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if pε < |A| < p1-ε, then always |A + A|+|A-1 + A-1| > pδ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution ( mod p) of certain rational functions on the primes (cf. [12]).

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