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Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a\in\mathbb{F}_{2}^{n}$ we have $f_{i}(a)=a_{i}$.

Can $f_{i}$'s be algebraically dependent over $\mathbb{F}_{2}$?.

Or even, can we say something about lower bound on the transcendence degree of this set $\{f_{1},f_{2},\ldots,f_{n}\}$ of polynomials over $\mathbb{F}_{2}$?

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$.

Can $f_{i}$'s be algebraically dependent over $\mathbb{F}_{2}$?.

Or even, can we say something about lower bound on the transcendence degree of this set $\{f_{1},f_{2},\ldots,f_{n}\}$ of polynomials over $\mathbb{F}_{2}$?