Let \(K\) be a field, fix an element \(\omega \in K\), and let \(n\) be a natural number. Write \(\{ 1,\omega \} ^n\) for the set of vectors \(x = (x_1,\dots ,x_n)\) with each coordinate \(x_i \in \{ 1,\omega \} \). Then for every function \(f : \{ 1,\omega \} ^n \to K\) there exist coefficients \(c_S \in K\), indexed by the subsets \(S \subseteq \{ 1,\dots ,n\} \), such that the squarefree (multilinear) polynomial

satisfies \(P(x) = f(x)\) for every \(x \in \{ 1,\omega \} ^n\).

Let \(K\) be a field and let \(\omega \in K\) satisfy \(\omega \ne 1\). Fix \(n, D \in \mathbb {N}\), and consider a monomial \(a\, X^m\) in the variables \(X_1,\dots ,X_n\) over \(K\), given by an exponent vector \(m : \mathrm{Fin}\, n \to \mathbb {N}\) and a coefficient \(a \in K\), whose total degree \(\sum _i m_i\) is at most \(D\). Then there exist coefficients \(c : \mathcal{S}_{n,D} \to K\), indexed by the squarefree supports \(s \subseteq \mathrm{Fin}\, n\) with \(\left\lvert s\right\rvert \le D\), such that the squarefree polynomial \(\sum _{\left\lvert s\right\rvert \le D} c(s)\prod _{i \in s} X_i\) takes the same value as \(a\, X^m\) at every point \(x\) of the root cube \(\{ 1,\omega \} ^n\).

Let \(K\) be a field and let \(\omega \in K\) satisfy \(\omega \ne 1\). For natural numbers \(n\) and \(D\), write \(\{ 1,\omega \} ^n\) for the set of vectors \(x : \mathrm{Fin}\, n \to K\) each of whose coordinates equals \(1\) or \(\omega \), and let \(\mathcal{S}_{n,D}\) be the collection of finsets \(s \subseteq \mathrm{Fin}\, n\) with \(\left\lvert s\right\rvert \le D\). Then for every polynomial \(Q \in K[X_i : i \in \mathrm{Fin}\, n]\) of total degree at most \(D\), there exist coefficients \(c : \mathcal{S}_{n,D} \to K\) such that the squarefree polynomial \(\sum _{\left\lvert s\right\rvert \le D} c(s)\, \prod _{i \in s} X_i\) agrees with \(Q\) at every point of \(\{ 1,\omega \} ^n\).

For natural numbers \(n\) and \(D\), let \(\mathcal{S}_{n,D}\) denote the (finite) type of low-degree squarefree supports on \(\mathrm{Fin}\, n\), that is, the subsets \(s \subseteq \mathrm{Fin}\, n\) with \(\left\lvert s\right\rvert \le D\). Then the number of such supports satisfies

Let \(\alpha \) and \(\beta \) be finite types, fix a function \(\mathrm{center} : \alpha \to \beta \), and let \(e \in \mathbb {N}\). Then the number of functions \(f : \alpha \to \beta \) that disagree with \(\mathrm{center}\) on at most \(e\) points of \(\alpha \) — that is, for which \(\left\lvert \{ a \in \alpha : \mathrm{center}(a) \ne f(a)\} \right\rvert \le e\) — satisfies

Let \(K\) be a finite field, let \(\omega \in K\), and let \(n, e \in \mathbb {N}\). Write \(\{ 1,\omega \} ^n\) for the root cube of vectors \(x : \{ 0,\dots ,n-1\} \to K\) each of whose coordinates equals \(1\) or \(\omega \), and consider functions \(f : \{ 1,\omega \} ^n \to K\), the Hamming distance between two such functions being the number of points of the cube on which they differ. For any function \(\mathrm{center} : \{ 1,\omega \} ^n \to K\), the number of functions whose Hamming distance from \(\mathrm{center}\) is at most \(e\) satisfies

where \(\left\lvert \{ 1,\omega \} ^n\right\rvert \) is the number of points of the cube and \(\left\lvert K\right\rvert \) is the number of elements of \(K\).

Let \(K\) be a finite field, let \(\omega \in K\) satisfy \(\omega \neq 1\), and fix natural numbers \(n, D, e, B\). Suppose the ball bound

holds, together with the strict counting inequality

Then it is not the case that every function \(f : \{ 1,\omega \} ^n \to K\) admits a multivariate polynomial \(Q \in K[X_i : i \in \mathrm{Fin}\, n]\) of total degree at most \(D\) that disagrees with \(f\) on at most \(e\) points of the cube \(\{ 1,\omega \} ^n\); that is, some \(f\) fails to be approximated in this sense by any such \(Q\).

Let \(K\) be a finite field and let \(\omega \in K\) satisfy \(\omega \neq 0\) and \(\omega \neq 1\), and fix natural numbers \(n, d, e, B\). Suppose that

and that

where \(\left\lvert \mathcal{S}_{n,\lfloor n/2\rfloor + d} \to K\right\rvert \) is the number of functions from \(\mathcal{S}_{n,\lfloor n/2\rfloor + d}\) to \(K\). Then there is no polynomial \(P \in K[X_1,\dots ,X_n]\) of total degree at most \(d\) whose bad count against the coordinate product \(x \mapsto \prod _{i} x_i\) on the root cube \(\{ 1,\omega \} ^{n}\) is at most \(e\); that is, every such \(P\) disagrees with \(\prod _i x_i\) at more than \(e\) points of \(\{ 1,\omega \} ^n\).