Good morning everyone.
My sure name is mokhtar and i am now reading a bout network coding, which demeand a lot of algebre and field theory, i have a little knowledge in this area, at now i have 3 theorem i haven't understanding.the first his name :
Sparse Zeros Lemma Let $f ∈ F_q[x_1, . . . , x_n]$ be a polynomial, not identically zero, whose $x_i$-degree is at most $d$ for all $i$. If $q > d$, then $f(a_1, . . . , a_n)$ not equal 0 for some $(a_1, . . . , a_n) ∈ F_n$.
can any one givme an example and the prove with alot of detail please,noting that in this theorem he use the following lemma :
Let $f ∈ F[x_1, . . . , x_n]$ have degree bounded by $d$ and let $A ⊆ F$ be a set of $d + 1$ distinct elements of F. If $f(a_1, . . . , a_n) = 0$ for all n-tuples $(a_1, . . . , a_n)$ in An, then $f ≡ 0$, i.e.,$f$ is identically the zero polynomial.
there is two more but i am interesting now in this