Computations and ML for surjective rational maps

2025/10/09 by Ilya Karzhemanov, Karzhemanov, Ilya
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cardinality (data modeling) #Computation #Endomorphism #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #Python (programming language) #Set (abstract data type) #Surjective function #Uniqueness #cs.LG #math.AG

paper · pdf · doi:10.48550/arxiv.2510.08093

openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

The present note studies surjective rational endomorphisms f: ℙ² \dashrightarrow ℙ² with cubic terms and the indeterminacy locus I_f ≠ ∅. We develop an experimental approach, based on some Python programming and Machine Learning, towards the classification of such maps; a couple of new explicit f is constructed in this way. We also prove (via pure projective geometry) that a general non-regular cubic endomorphism f of ℙ² is surjective if and only if the set I_f has cardinality at least 3.

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