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Injective endomorphisms of real algebraic sets are surjective. (English) Zbl 0933.14036

It is proved that injective algebraic mappings from a real algebraic set \(X\) to itself are surjective. In the complex case (more generally over algebraically over algebraically closed field of characteristic 0) and for regular mappings this result was proved by J. Ax [Ann. Math., II. Ser. 88, 239-271 (1968; Zbl 0195.05701)]. In the real case this result was proved by A. Białynicki-Birula and M. Rosenlicht [Proc. Am. Math. Soc. 13, 200-203 (1962; Zbl 0107.14602)] for \(X=\mathbb{R}^n\), later by A. Borel [Arch. Math. 20, 531-537 (1969; Zbl 0189.21402)] for \(X\) smooth. A. Tognoli [Boll. Unione Mat. Ital., VII. Ser., B7, No.3, 719-733 (1993; Zbl 0803.14028)] proposed a proof in the case where \(X\subset \mathbb{R}^n\) may be singular, buthis proof is incomplete. The precise statement of the result proved by the author is the following:
If \(X\subset \mathbb{R}^n\) is algebraic, \(f:X\to X\) is an injective continuous mapping such that the graph of \(f\) is algebraic (in particular \(f\) may be regular), then \(f\) is surjective. The proof of the result, in somewhat more general setting, uses Borel-Moore homology for semialgebraic sets. The essential ingredient is the notion of arcwise symmetric set, introduced by the author [K. Kurdyka, Math. Ann. 282, No. 3, 445-462 (1988; Zbl 0686.14027)] and generalized in the paper under review. It allows to define a noetherian topology, finer than the Zariski topology, on any semialgebraic set. Irreducible components (in this topology) of sets or germs are preserved by injective algebraic mappings.

MSC:

14P10 Semialgebraic sets and related spaces
14A10 Varieties and morphisms
32C05 Real-analytic manifolds, real-analytic spaces
32C07 Real-analytic sets, complex Nash functions
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