Estimating the number of real zeros of linear combinations of radicals of polynomials
arXiv:2609.02871
2026
Architecture
1 ideas extracted · analyzed Sep 3, 2026
What the math gives to ML
The paper provides explicit oscillation bounds for linear combinations of algebraic-radical basis functions. Its transferable asset is an architecture-level complexity certificate: a nonzero sum of n square roots of positive quadratics has at most 2n real zeros, while a Wronskian argument gives polynomial zero bounds for more general radical families. This suggests replacing highly oscillatory Fourier or sinusoidal coordinate features in one-dimensional neural fields with trainable radical features whose sign changes are structurally controlled. The most credible use is certified low-complexity implicit functions, monotone or piecewise-smooth regression, and anti-ringing coordinate networks rather than general-purpose language models.
Ideas from this paper
Unverified
2026
Replace Fourier or sinusoidal one-dimensional coordinate features with a trainable radical layer f(x)=sum_i c_i sqrt(P_i(x)), where every P_i is a strictly positive quadratic. A nonzero scalar output formed by such a layer has at most 2n distinct real zeros, providing an explicit bound on sign changes and suppressing uncontrolled ringing. Use the radical features as an input embedding for a conventional MLP or neural implicit field.
Useful5/10
Difficulty4/10
Novelty8/10