Research proof index

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.

Lower bound: (gs_degree_bound + 1)^2 > (m+1/2)^2 * k * n.

Divisibility via the rate-corrected GS system. Uses gs_degree_bound (ρ=k/n) and gs_johnson instead of the conservative proximity_gap versions.