Are all multivariate functions on finite fields equivalent to a unique polynomial of smallest degree?

Take, for example $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$, $n=2$, and $f(x, y) = xy$. It is easy to check that no polynomial of degree $\leq 1$ agrees with this function on all inputs, i.e., the minimal degree is $2$. But $xy + x(x-1)$ is another polynomial of degree $2$ which agrees with the function on all inputs.

A similar example can be constructed whenever $n\geq |\mathbb{F}|$.

Edit: actually, $n=2$ seems to be enough for any finite field: take $p(x,y) = \left(\prod\limits_{a\in\mathbb{F}\backslash \{0\}} (x-a)\right) y$ and $q(x,y) = p(x,y) + \prod\limits_{a\in\mathbb{F}} (x-a)$.

We assume that the field $\Bbb F$ is finite and $|\Bbb F|=q$. Litho’s example shows that it can happen that $|L|>1$.

On the other hand, we can achieve an uniqueness of polynomials of $L$, imposing a natural restriction on their degrees. Indeed, given $f$, by induction with respect to $n$ we can construct a multidimensional Lagrange interpolation polynomial for $f$, which has degree at most $q-1$ with respect to each variable (and so a total degree at most $(q-1)n$). It follows that the set $L$ is non-empty.

Since $x^q=x$ for each $x\in\mathbb F$, given any polynomial $p\in L$ represented as a sum of monomials, if we substitute, as orangeskid suggested, in each of the monomials a factor $x_i^{n_i}$ by $x_i^{m_i}$, where $m_i\in \{1,2,\ldots, q-1\}$, and $n_i\equiv m_i \mod (q-1)$, we obtain a reduced polynomial $\bar p$ which has degree at most $q-1$ with respect to each variable and $\bar p(x)=p(x)$ for each $x\in \Bbb F^n$.

For any polynomials $p,r\in L$, a polynomial $\bar p-\bar r$ has degree at most $q-1$ with respect to each variable. So it is zero by the following

Theorem (Combinatorial Nullstellensatz II). [A] Let $\Bbb F$ be a field and $f\in \Bbb F[x_1,\dots, x_n]$. Suppose $\deg f =\sum_{i=1}^n t_i$ for some nonnegative integers $t_i$ and the coefficient of $\prod_{i=1}^n x_i^{t_i}$ is nonzero. If $S_1,\dots, S_n\subset \Bbb F$ such that $|S_i| > t_i$ then there exists $s_1\in S_1,\dots, s_n\in S_n$ such that $f(s_1,\dots,s_n)\ne 0$.

References

[A] N. Alon, Combinatorial Nullstellensatz, Combinatorics, Probability and Computing 8 (1999), 7–29.

See (3) in this answer for more references.