Proofs of Two Conjectures on Ternary Weakly Regular Bent Functions

In this paper, we study ternary monomial functions of the form
f
(
x
) = Tr
n
(
axd
), where
x
isin \BBF
₃

n
and
Trn
: \BBF
₃

n
rarr \BBF
₃
is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising in the 2006
IEEE Transactions on Information Theory
paper (vol. 52, pp. 2018-2032, 2006) are weakly regular bent, thus settling a conjecture of Helleseth and Kholosha. We also prove that the Coulter-Matthews bent functions are weakly regular.