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I am a mathematical physicist and I am studying certain discrete dynamical systems defined in terms of piecewise linear mappings, which may be expressed in terms of expressions over the max-plus semi-field.

In the study of the points of non-differentiability of these piecewise linear mappings, I was wondering, is there an established proof that (in n-dimensions) every set of points of non-differentiability of a piecewise linear mapping (with rational slopes) may be expressed as the image of a variety over a non-archimedean valuation field under the endowed valuation?

Is there some, possibly constructive proof, with some sort of standard construction?

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  • $\begingroup$ Could you name your spaces, so I can understand what you are asking? Right now, it looks like you start with a PL map from $\mathbb{R}^n$ to itself, and are considering the subspace $X$ of the domain on which the map is non-differentiable. Am I correct that you want to know if $X$ is the image under the valuation map of some variety in $\mathbb{A}^n_K$, where $K$ is a field endowed with a nonarchimedean real valuation? $\endgroup$
    – S. Carnahan ♦
    Oct 19, 2010 at 3:35
  • $\begingroup$ Indeed, you are correct. I have $X \subset \mathbb{R}^n$ which is the set of points in which a PL mapping, $f : \mathbb{R}^n \to \mathbb{R}^m$, is not differentiable. One may replace $\mathbb{R}$ with $S = (\mathbb{R} \cup \{\infty\}, \max, + )$. Now, the field that seems popular would be letting $\mathbb{K}$ be the set of algebraic functions (with usual index map). Not being an expert in the field, I would like to know if it has been established that $X$ is the image variety over $\mathbb{A}_{\mathbb{K}}^n$. Is there, for example, a standard construction of an underlying ideal? $\endgroup$
    – Chris Ormerod
    Oct 22, 2010 at 1:04
  • $\begingroup$ I would say that the corner locus of your map is the union of corner loci for the restriction to each coordinate of $R^m$. Therefore it is just a union of tropical varieties. $\endgroup$
    – Nikita Kalinin
    Aug 5, 2017 at 20:52

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