Nonrecursive functions in real algebraic geometry. (English) Zbl 0692.14013
One could easily think that all the questions arising from real algebraic geometry have a constructive answer: The author shows here that this is not always the case. A function is build up which cannot be bounded by a recursive function. This is a consequence of a theorem of Novikov [see I. A. Volodin, V. E. Kuznetsov, A. T. Fomenko, Russ. Math. Surv. 29, No.5, 71-172 (1974); translation from Usp. Mat. Nauk 29, No.5(179), 71-168 (1974; Zbl 0303.57002)] about recognising algorithmically whether smooth manifolds are diffeomorphic to the standard sphere or not. Novikov’s result is deduced from another one about non decidability for finitely presented groups.
The function is constructed as follows: one considers all the compact algebraic hypersurfaces of degree \( d\) in \({\mathbb{R}}^{n+1}\) which are isotopic to the standard sphere \(S^ n\); one can choose an isotopy passing only through algebraic hypersurfaces. These ones have a degree which is bounded by an \(integer\quad D\) depending on n and d. The function we looked for is precisely, for fixed d, the map \(n\to \inf (D(n,d))\).
The function is constructed as follows: one considers all the compact algebraic hypersurfaces of degree \( d\) in \({\mathbb{R}}^{n+1}\) which are isotopic to the standard sphere \(S^ n\); one can choose an isotopy passing only through algebraic hypersurfaces. These ones have a degree which is bounded by an \(integer\quad D\) depending on n and d. The function we looked for is precisely, for fixed d, the map \(n\to \inf (D(n,d))\).
Reviewer: F.Broglia
MSC:
| 14Pxx | Real algebraic and real-analytic geometry |
| 03D15 | Complexity of computation (including implicit computational complexity) |
| 14M10 | Complete intersections |
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