Suppose H has degree at most d_H and Q is a positive integer. Set q=floor(d_H/Q) and J=q+1. At each t=1,...,Q obtain the coefficients of z^0,...,z^(J-1) in H(t+z). These QJ numbers determine H because QJ>d_H. If two candidates share them, their difference is divisible by (T-t)^J at each distinct t, hence by their product of degree QJ. Such a difference must be zero. Solving the resulting rational linear equations recovers H(0), the count. Example: for a quadratic, Q=2 and J=2 give two values and two first-order coefficients; the fourth datum is redundant but harmless when consistent.