Upper bound for the length of functions over a finite field in the class of pseudopolynomials

An exclusive-OR sum of pseudoproducts (ESPP), or a pseudopolynomial over a finite field is a sum of products of linear functions. The length of an ESPP is defined as the number of its pairwise distinct summands. The length of a function $f$ over this field in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function $L_k^{\text{ESPP}}(n)$ on the set of functions over a finite field of $k$ elements in the class of ESPPs is considered; it is defined as the maximum length of a function of n variables over this field in the class of ESPPs. It is proved that $L_k^{\text{ESPP}}(n)=O(k^n/n^2)$.