On two conjectures about the intersection distribution

DOI10.1007/S10801-021-01095-XarXiv2010.00312WikidataQ123137528 ScholiaQ123137528MaRDI QIDQ6350300

Publication date: 1 October 2020

Abstract: Recently, S. Li and A. Pottcite{LP} proposed a new concept of intersection distribution concerning the interaction between the graph

(x, f (x)) | x i n F_{q}

of

f

and the lines in the classical affine plane

A G (2, q)

. Later, G. Kyureghyan, et al.cite{KLP} proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over

F_{q}

with

q

both odd and even. They also proposed several conjectures in cite{KLP}. In this paper, we completely solve two conjectures in cite{KLP}. Namely, we prove two classes of power functions having intersection distribution:

v_{0} (f) = f r a c q (q - 1) 3, v_{1} (f) = f r a c q (q + 1) 2, v_{2} (f) = 0, v_{3} (f) = f r a c q (q - 1) 6

. We mainly make use of the multivariate method and QM-equivalence on

2

-to-

1

mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations.

Mathematics Subject Classification ID

Finite affine and projective planes (geometric aspects) (51E15) Polynomials over finite fields (11T06)

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