On the curvature of functions over finite fields

The curvature $\sigma(f)$ of a function $f$ over the finite field $P=\text{GF}(q)$ is defined as the sum of the modules of the coefficients of the function's expansion in the basis of characters. In this paper, an estimate of the curvature is obtained. It is proven that for a function $f: P^n\rightarrow P$ in $n$ variables the following holds: $1\le\sigma(f)\le q^{{n}/{2}}$. The conditions for attainability of estimates have been established. Also, the curvature of various classes of functions over finite field is investigated. Let $g(x_1,\ldots,x_n) = \pi(x_1) + f(x_2,\ldots,x_n)$, where $f:P^{n-1}\rightarrow P$ and $\pi$ is a permutation defined by a polynomial $\pi(x) = x^k + c_{k-1}x^{k-1} + \ldots+c_1x+c_0$, $(k,p)=1$, $c_i\in P$, $i\in \{0,\ldots, k-1\}$. It is shown that $\sigma(g)=\sigma(f)$ for $k=1$ and $\sigma(g)\le\sigma(f)(k-1)(q-1)/\sqrt{q}$ for $k>1$. For the function $g(x_1, \ldots, x_n)=x_n^{q-1}f_1(x_1,\ldots, x_{n-1})+(1-x_n^{q-1})f_2(x_1,\ldots, x_{n-1})$, it is proven that $\sigma(g) \le (2q-2)\sigma(f_1)/q+\sigma(f_2)$.