On the Evaluation of Weil Sums of Dembowski–Ostrom Polynomials

✍ Scribed by Donald Mills

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 117 KB
• Volume: 92
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2701

✦ Synopsis

Let F q denote the finite field of q elements, q=p e odd, let q 1 denote the canonical additive character of F q where q 1 (c)=e 2piTr(c)/p for all c ¥ F q , and let Tr represent the trace function from F q to F p . We are interested in evaluating Weil sums of the form

Coulter has determined the value of S when D(x)=ax p a +1 ; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of F l q -solutions to the diagonal-type equation ; l i=1 x p c +1 i +(x i +l) p c +1 =0, where l is even, e/gcd(c, e) is odd, and h(X)=l p e-c

X p e-c +l p c X is a permutation polynomial over F q .