arXiv:2607.01572 2026 Architecture 1 ideas extracted · analyzed Aug 30, 2026
The paper gives an explicit positive triangular basis transformation that preserves real-rootedness for polynomials with nonnegative coefficients. This is a certified stability-preserving coordinate change, rather than an ordinary feature map: a polynomial whose roots are all real and nonpositive remains real-rooted after transformation into the Narayana basis. The most plausible neural-network transfer is a structured polynomial feature layer whose coefficients are generated from nonnegative root parameters and then transformed by a fixed Narayana matrix, with root geometry used as a hard architectural constraint.
Replace a monomial polynomial feature block by a fixed Narayana basis transformation. If the input polynomial has nonnegative coefficients and only real roots, the transformed polynomial is guaranteed to have only real roots as well, giving a certified stability-preserving coordinate change for polynomial neural networks.