\({\mathbb {F}_{q}}\)

the finite field with \(q\) elements;

\(\mathrm{Map}(X,{\mathbb {F}_{q}})\)

the vector space of all functions \(f : X \to {\mathbb {F}_{q}}\) for a given set \(X\);

\(\mathrm{Z}(f_{1}, \ldots , f_{k})\)

the set of common zeros of \(f_{1}, \ldots , f_{k} \in \mathrm{Map}(X,{\mathbb {F}_{q}})\);

\(\mathrm{Span}(f_{1}, \ldots , f_{k})\)

the subspace of \(\mathrm{Map}(X,{\mathbb {F}_{q}})\) generated by \(f_{1}, \ldots , f_{k}\);

\(\mathbb {A}^{n}(\mathbb {K})\)

the affine \(n\)-space over a field \(\mathbb {K}\);

\(\mathbb {P}^{n}(\mathbb {K})\)

the projective \(n\)-space over a field \(\mathbb {K}\);

\(\left[\alpha _{1} \colon \ldots \colon \alpha _{n+1} \right]\)

a set of homogeneous coordinates for a point in \(\mathbb {P}^{n}(\mathbb {K})\).