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This work explores gaps in the number of non-vanishing components of linear systems from rational sphere maps.

April 26, 2026International Journal of Mathematics0 citations

Values of ℓ 0 Norms for Solutions of Linear systems Arising From Monomial Sphere Maps

JDJohn P. D’AngeloUniversity of Illinois Urbana-Champaign

Key Points

This work explores gaps in the number of non-vanishing components of linear systems from rational sphere maps.
Combinatorial approaches are employed to derive explicit formulas.
An integer is identified for minimum embedding dimensions of polynomial sphere maps.
A conjecture regarding sharpness of the derived integer is proposed.
An explicit formula for integer values exceeding a certain threshold is established.
The asymptotic value found improves upon previous quadratic suggestions.
The conjectured value is stated to be sharp, though unproven.

Abstract

We consider gaps that arise in the possible numbers of non-vanishing components for certain linear systems that arise when studying rational sphere maps. When the source dimension is Formula: see text, we find an explicit formula for an integer Formula: see text such that every number exceeding Formula: see text is the value for the minimum embedding dimension of some polynomial (hence rational) sphere map. We conjecture, but do not prove, that Formula: see text is sharp. This number is asymptotic to Formula: see text; this asymptotic value improves what has been suggested to be quadratic in Formula: see text. Our methods are combinatorial.

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