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Author

Lek-Heng Lim

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Preprint Sep 2026

Pierce-Birkhoff conjecture is false

We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our...

Ze-Hua Lai, Lek-Heng Lim, Jun-Yu Ren · 0 citations
#machine learning Jun 2026

Low-dimensional topology of deep neural networks

The results suggest that low-dimensional topology can be a useful tool to guide designs of AI architectures, and generalize the results from $d = 3$ to arbitrary $d>3$.

Jun Ren, Lek-Heng Lim · 0 citations

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