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September 17, 2025Open Access

A degree bound for planar functions

Key Points

The algebraic degree of composed functions shows a bound tied to planar functions.
For a finite field with characteristic greater than 5, the complete classification of planar monomials is achieved.
Stickelberger's theorem provides a crucial foundation for establishing the degree bound in this context.
The classification ties into insights involving the digit sum of integers modulo q-1, revealing additional properties.

Abstract

Using Stickelberger's theorem on Gauss sums, we show that if \ (F\) is a planar function on a finite field \ (Fq\), then for all non-zero functions \ (G: Fq Fq\), we have equation* d₀₋₆ (G F) - d₀₋₆ (G) n (p-1) 2, equation* where \ (q = pⁿ\) with \ (p\) a prime and \ (n\) a positive integer, and \ (d₀₋₆ (F) \) is the algebraic degree of \ (F\), i. e. , the maximum degree of the corresponding system of \ (n\) lowest-degree interpolating polynomials for \ (F\) considered as a function on \ (Fₚⁿ\). This bound implies the (known) classification of planar polynomials over \ (Fₚ\) and planar monomials over \ (F₏ℂ\). As a new result, using the same degree bound, we complete the classification of planar monomials for all \ (n = 2ᵏ\) with \ (p›5\) and \ (k\) a non-negative integer. Finally, we state a conjecture on the sum of the base-\ (p\) digits of integers modulo \ (q-1\) that implies the complete classification of planar monomials over finite fields of characteristic \ (p›5\). Mathematics Subject Classifications: 05B25, 11T06, 11T24Keywords: Planar function, algebraic degree, Stickelberger's theorem, digit sum

A degree bound for planar functions

Key Points

Abstract

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