F. Sottile, Enumerative geometry for real varieties, Algebraic Geometry, Santa Cruz 1995. Home/Search Document Details and Download
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Algebraic Geometry and Computer Vision: Polynomial Systems, Real .. - Petitjean (1999) (1 citation) (Correct)
....can be real. For instance, 83] investigated the number of conics tangent to ve general conics in the real case. They proved that in fact all (i.e. 3264 as was found by de Jonqui eres in 1859, and again by Chasles in 1864 see [116] can be real. In a recent series of papers, F. Sottile [95, 96, 97] was the rst to give a general statement in this direction: he produces large classes of fully real non trivial enumerative problems. For instance, he proves in [96] that for any problem of enumerating lines in incident on real linear subspaces in general position, all solutions can be real. He ....
....subspaces in general position, all solutions can be real. He later expended this result to other classes of enumerative problems by giving a procedure to create new fully real problems from existing ones [97] Despite these positive results , some negative ones have also popped up. Sottile [95] notes interestingly that these negative examples all seem to involve intersections of non general subvarieties (i.e. they do not intersect in the way two random varieties would intersect) For instance, F. Klein [51] showed that at most n(n 2) of the 3n(n 2) exes on a real plane curve of degree ....
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F. Sottile. Enumerative geometry for real varieties. In Jnos Kollr, editor, Algebraic Geometry, Santa Cruz 1995, volume 61, number 1 of Proceedings and Symposia in Pure Mathematics, pages 435-447. American Mathematical Society, 1997.
Felix Klein's Paper On Real Flexes Vindicated - Ronga (1997) (1 citation) (Correct)
....in algebraic geometry, the number of solutions of an enumerative problem over the reals is bounded by the number of solutions over the complex. So, two natural questions arise : 1) Is it possible to arrange that all the solutions are real If it is the case, the problem is called fully real in [6]. 2) Which intermediate number of solutions can be obtained Examples. i) According to B ezout s theorem, two plane curves in general position, of degree m and n respectively intersect in m Delta n points. By taking curves consisting of lines, some of them real, some pairwise complex ....
F. Sottile, Enumerative Geometry for real Varieties, to appear.
Lines Tangent to 2n-2 Spheres in R^n - Sottile, Theobald (2002) Self-citation (Sottile) (Correct)
....tangent lines. Moreover, this configuration can be achieved with unit spheres. Thus the bound for real common tangents equals the (a priori greater) bound for complex common tangents; so this problem of common tangents to spheres is fully real in the sense of enumerative real algebraic geometry [15, 16]. We prove Statement (a) in Section 2 and Statement (b) in Section 3, where we explicitly describe configurations with 3.2 n 1 common real tangents. Figure i shows a configuration of 4 spheres in ]a with 12 common tangents (as given in [9] FIGURE 1. Spheres with 12 real common tangents In ....
F. Sottile, Enumerative geometry for real varieties, Algebraic Geometry, Santa Cruz 1995.
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F.Sottile, Enumerative geometry for real varieties, Proc. Sympos. Pure Math (1997), 435-447.
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