  In the last decade, the approximate basis computation of vanishing ideals has
been studied extensively in computational algebra and data-driven applications
such as machine learning. However, symbolic computation and the dependency on
term order remain essential gaps between the two fields. In this study, we
present the first $\textit{monomial-agnostic}$ basis computation, which works
fully numerically with proper normalization and without term order. This is
realized by gradient normalization, a newly proposed data-dependent
normalization that normalizes a polynomial with the magnitude of gradients at
given points. The data-dependent nature of gradient normalization brings
various significant advantages: i) efficient resolution of the spurious
vanishing problem, the scale-variance issue of approximately vanishing
polynomials, without accessing coefficients of terms, ii) scaling-consistent
basis computation, ensuring that input scaling does not lead to an essential
change in the output, and iii) robustness against input perturbations, where
the upper bound of error is determined only by the magnitude of the
perturbations. Existing studies did not achieve any of these. As further
applications of gradient information, we propose a monomial-agnostic basis
reduction method and a regularization method to manage positive-dimensional
ideals.
