Vandermonde polynomial initialization

Integrate this at initialization of a scalar polynomial layer whose hidden units compute h_j(x)=(w_jx+b_j)^d and whose output is y=c+\sum_jv_jh_j(x). The simplest stable construction sets every w_j=1, chooses d pivots b_j on a bounded interval such as equally spaced points in [-1,1], and solves the Vandermonde linear system before training. Given desired polynomial coefficients p_0,\ldots,p_d, construct the matrix V[i,j]=b_j^i for i=0,\ldots,d-1, solve V a=r with r_i=p_{d-i}/\binom{d}{i}, set v_j=a_j, and set c=p_0-\sum_j a_jb_j^d. Use QR or SVD rather than an explicit inverse; rescale x into [-1,1] first because equally spaced Vandermonde systems become ill-conditioned at large degree. For a target affine map, use p_0 and p_1 from a linear regression and set p_k=0 for k\ge2. Optionally freeze this layer for a warm-up period, then unfreeze all parameters. What is computed analytically is the exact coefficient synthesis; what is estimated empirically is the fitted polynomial coefficients and whether later optimization preserves useful conditioning. First test d=3,5,7 on a one-dimensional regression benchmark such as fitting sin(x) on [-1,1], comparing random Xavier initialization against Vandermonde initialization at equal width and optimizer settings. Measure initial function error, loss after a fixed number of updates, gradient norms, and frequency of divergence. Success is lower initial loss and faster loss reduction without large output-weight explosions.