Research proof index

The coefficient of X^i Y^j in a linear combination of monomials is the coefficient of the combination.

The degree of Q(X, P(X)) is bounded by the (1, k-1)-weighted degree of Q, provided deg(P) ≤ k - 1.

If Q has high multiplicity at (0,0) (meaning all coefficients c_{i,j} with i+j < m are zero) and P(0)=0, then Q(X, P(X)) is divisible by X^m.

If $Q$ satisfies the weighted degree bound and vanishes to order $m$ at each point $(\omega_i, f_i)$, and if $P$ is a codeword close enough to $f$, then $Y - P(X)$ divides $Q(X,Y)$.

There exists a non-zero polynomial satisfying the conditions.

Generalized existence: non-zero kernel element for arbitrary degree bound D, given numVars k D > numConstraints n m.