1 This is a Test
In this chapter, this is a test.
If there is a polynomial \(f\) with degree at most \(k-1\) such that \(\Delta (f,w)\leq e\), then there exists \(E\) and \(Q\) satisfying:
\(\textrm{deg}(E(X))=e\) and \(E(X)\) is monic.
\(\textrm{deg}(Q(X))\leq e+k-1\).
\(w_i\cdot E(\alpha _i)=Q(\alpha _i)\) for all \(i=1,...,n\).
Proof
Consider the error-locator polynomial of the form
\[ E(X)=\Pi _{i: f(a_i)\neq y_i}(x-a_i). \]
If \(S\) is a finite set, and \(\sum _{s \in S} w_s = 1\) for some non-negative \(w_s\), and \(p_s \in [0,1]\) for all \(s \in S\), then
\[ \sum _{s \in S} w_s h(p_s) \leq h(\sum _{s \in S} w_s p_s). \]