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Rob Eggermont

Publications and source records attributed to Rob Eggermont.

5 recordsLinked to original sources
arXiv2026

Improved unirationality for GL-varieties

A $\mathbf{GL}$-variety is a typically infinite dimensional variety equipped with a suitable action of the infinite general linear group $\mathbf{GL}$. In earlier work, we established the unirationality theorem: an irreducible $\mathbf{GL}$-variety admits a dominant map from a particularly simple $\mathbf{GL}$-variety, namely, the product of an irreducible finite-dimensional variety with trivial $\mathbf{GL}$-action and an infinite-dimensional affine space on which $\mathbf{GL}$ acts linearly. The main result of this paper states that this map can in fact be constructed to be surjective rather than merely dominant. An immediate application is that secant varieties to varieties of tensors, which are typically constructed as image closures of certain $\mathbf{GL}$-equivariant maps, are in fact also images of (more complicated) $\mathbf{GL}$-equivariant maps. We derive several consequences of this improved unirationality theorem.

math.AG↗
arXiv2025

Global Positioning on Earth

Contrary to popular belief, the global positioning problem on earth may have more than one solutions even if the user position is restricted to a sphere. With 3 satellites, we show that there can be up to 4 solutions on a sphere. With 4 or more satellites, we show that, for any pair of points on a sphere, there is a family of hyperboloids of revolution such that if the satellites are placed on one sheet of one of these hyperboloid, then the global positioning problem has both points as solutions. We give solution methods that yield the correct number of solutions on/near a sphere.

math.MG↗
arXiv2024

Topological Noetherianity of the infinite half-spin representations

We prove that the infinite half-spin representations are topologically Noetherian with respect to the infinite spin group. As a consequence we obtain that half-spin varieties, which we introduce, are defined by the pullback of equations at a finite level. The main example for such varieties is the infinite isotropic Grassmannian in its spinor embedding, for which we explicitly determine its defining equations.

math.AG↗
arXiv2023

Uniformity for limits of tensors

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the locus with "border rank" $\le r$) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank $\le r$ do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on $\mathbf{GL}$-varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on $\mathbf{GL}$-varieties.

math.AG↗