Approximate basis computation of vanishing ideals has recently 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 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. Its data-dependent nature brings various 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.
We determine the conjugacy class fusion from certain maximal subgroups of the Monster to the Monster, to justify the addition of these data to the Character Table Library in the computational algebra system GAP. The maximal subgroups in question are , , , and . Our proofs are supported by reproducible calculations carried out using the Python package mmgroup, a computational construction of the Monster recently developed by Seysen.
A Howe curve is defined as the normalization of the fiber product over a projective line of two hyperelliptic curves. Howe curves are very useful to produce important classes of curves over fields of positive characteristic, e.g., maximal, superspecial, or supersingular ones. Determining their feasible equations explicitly is a basic problem, and it has been solved in the hyperelliptic case and in the non-hyperelliptic case with genus not greater than 4. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus 5. We also determine the number and type of singularities on our sextic model, and prove that the singularities are generically 4 double points. Our results together with Moriya-Kudo's recent ones imply that for each , there exists a non-hyperelliptic curve H of genus 5 with such that its associated plane sextic has s double points.
In algebraic geometry or number theory, enumerating or finding superspecial curves in positive characteristic p is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group. By executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for all primes p with , and in finding a single one for all primes p with .
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