Summary: Let \(p\) be a prime number and \(F=\text{GF}(p)\). Suppose \(V_n\) is an \(n\)-dimensional vector space over \(F\) and \(e\) is a basis of \(V_n\). Also, let \(\varphi\colon V_n\to F\). The function \(\varphi\) is called \(e\)-homogeneous if \(\varphi(x)=\pi_{\varphi,e}(\mathbf{x})\) for all \(x\in V_n\), where \(\pi_{\varphi,e}\) is an \(n\)-variate homogeneous polynomial over \(F\) of degree at most \(p-1\) in each variable and \(\mathbf{x}\) is the coordinate vector of \(x\) with respect to the basis \(e\). The function \(\varphi\) is said to be nondegenerate if \(\deg\varphi\ge1\) and \(\deg\partial_v\varphi=(\deg\varphi)-1\) for any \(v\in V_n\setminus\{0\} \), where \((\partial_v\varphi)(x)=\varphi(x+v)-\varphi(x)\) for all \(v,x\in V_n\). This notion was introduced by O. A. Logachev, A. A. Sal’nikov, and V. V. Yashchenko in the case when \(p=2\). Our main results are as follows. First, we obtain a formula for the number \(\text{HN}_p(n,d)\) of \(e\)-homogeneous nondegenerate functions \(\varphi\colon V_n\to F\) of degree \(d\) (this number does not depend on \(e)\). Namely, if \(n\ge1\) and \(d\in\{1,\dots,n(p-1)\} \), then \(\text{HN}_p(n,d)=\sum_{k=0}^n(-1)^k p^{\binom{k}{2}+\genfrac{\{}{\}}{0pt}{}{n-k}{d}_p} \begin{bmatrix}n \\ k\end{bmatrix}_p=\sum_{S\subseteq\{1,\dots,n\}} (-1)^{|S|}p^{\sigma(S)-|S|+\genfrac{\{}{\}}{0pt}{}{n-|S|}{d}_p}\), where \(\genfrac{\{}{\}}{0pt}{}{m}{d}_p\) is the generalized binomial coefficient of order \(p\), \(\begin{bmatrix} n \\ k \end{bmatrix}_p\) is the Gaussian binomial coefficient, and \(\sigma(S)\) is the sum of all elements of \(S\). The proof of this formula is based on the Möbius inversion. Previously, only formulas for \(\text{HN}_p(n,2)\) were known; unlike our formula, their forms depend on the parities of \(p\) and \(n\). Second, we prove that \(\text{HN}_p(n,d)\ge p^{\genfrac{\{}{\}}{0pt}{}{n}{d}_p} -1-(p^n-1)\left(p^{\genfrac{\{}{\}}{0pt}{}{n-1}{d}_p}-1\right)/(p-1)\) for any \(d\ge1\) and \(n\ge d/(p-1)\). Using this bound, we obtain that if \(d\ge3\), then \(\text{HN}_p(n,d)\sim p^{\genfrac{\{}{\}}{0pt}{}{n}{d}_p}\) as \(n\to\infty \). For \(p=2\) the last two statements were proved by Yu. V. Kuznetsov. The proofs of our main results use a Jennings basis of the group algebra \(FG_n\), where \(G_n\) is an elementary abelian \(p\)-group of rank \(n\).