Frank Sottile: Papers in Real Algebraic Geometry

Papers in Combinatorics,   Geometry,   Schubert calculus,   Combinatorial Hopf Algebras,   Computational Algebraic Geometry,   Applicable Algebraic Geometry,   and   Computational Geometry.

1	Betti number bounds for fewnomial hypersurfaces via stratified Morse theory, with Frédéric Bihan, 7 pages. arXiv:0801.2554.
2	Bounds on the number of real solutions to polynomial equations, with Daniel J. Bates, Frédéric Bihan, IMRN, 2007, 2007:rnm114-7.
3	Gale duality for complete intersections, with Frédéric Bihan, Annales de l'Institut Fourier, Tome 58 (2008) fasicule 3, pp.~877-891. A Singular script which computes examples of the Kouchnirenko Theorem for systems of Master functions.
4	Linear precision for parametric patches with Luis Garcia 20 pages. arXiv:0706.2116.
5	Sharpness of fewnomial bound and the number of components of a fewnomial hypersurface, with Frédéric Bihan and J. Maurice Rojas. IMA Volume 146: Algorithms in Algebraic Geometry edited by Alicia Dickenstein, Frank-Olaf Schreyer, and Andrew J. Sommese. pp. 15--20, Springer, New York, 2007.
6	Line problems in nonlinear computational geometry, with Thorsten Theobald. Surveys on Discrete and Computational Geometry - Twenty Years Later, Contemporary Mathematics, 453 AMS, 2008, 411--432.
7	New fewnomial upper bounds from Gale dual polynomial systems, with Frédéric Bihan, Moscow Mathematics Journal, 7 (2007), Number 3, 387--407.
8	Real Hessian Curves with Adriana Ortiz-Rodríguez. Boletín de la Sociedad Mathemática Mexicana, Volume 13 Number 3 (2007), 157--166. Companion web page.
9	Lines tangent to four triangles in three-dimensional space with Hervé Brönnimann, Olivier Devillers, and Sylvain Lazard. Discrete and Computational Geometry, 37, No. 3, (2007), 369--380. Companion web page.
10	Experimentation and conjectures in the real Schubert calculus for flag manifolds, with James Ruffo, Yuval Sivan, Evgenia Soprunova, Experimental Mathematics, 15, No. 2 (2006), 199-221.
11	Polynomial systems with few real zeroes, with Benoît Bertrand and Frédéric Bihan, Mathematisches Zeitschrift, 253 (2006), no. 2, 361--385.
12	Lower Bounds for Real Solutions to Sparse Polynomial Systems, with Evgenia Soprunova, Advances in Mathematics, Volume 204, Issue 1, 1 August 2006, Pages 116-151.
13	Transversals to line segments in R3, with Hervé Brönnimann, Hazel Everett, Sylvain Lazard, and Sue Whitesides, Discrete and Computational Geometry, Volume 34, Number 3, (2005), 381 -- 390. Companion web page.
14	Real k-flats tangent to quadrics in Rn, with Thorsten Theobald. Proc. Amer. Math. Soc., 133 (2005), 2835-2844.
15	The envelope of lines meeting a fixed line and tangent to two spheres, with Gábor Megyesi, Discrete and Computational Geometry, 33, Number 4, (2005) 617--644.     Companion web page.
16	A New Approach to Hilbert's Theorem on Ternary Quartics, with Victoria Powers, Bruce Reznick, and Claus Scheiderer, Comptes Rendus Mathematique (Paris), 339, Issue 9, (2004), 617--620.
17	Maximally inflected real rational curves, with Viatcheslav Kharlamov. Moscow Mathematics Journal 3 (2003), no. 3, 947--987, 1199--1200.     Companion Web Page.
18	Toric ideals, real toric varieties, and the moment map, in Topics in Algebraic Geometry and Geometric Modeling, ed. by R. Goldman and R. Krasuaskas, Contemp. Math. 334, 2003. pp. 225-240. (Proceedings of AGGM, Vilnius, Lithuania.)
19	Common transversals and tangents to two lines and two quadrics in P3, with Gábor Megyesi and Thorsten Theobald. Discrete and Computational Geometry, 30, (2003), pp. 543-571.     Companion Web Page.
20	Enumerative Real Algebraic Geometry, in Algorithmic and Quantitative Aspects of Real Algbraic Geometry, S. Basu and L. Gonzalez-Vega, eds., DIMACS series 60, AMS, 2003. pp. 139--180.       Expanded on-line version.
21	Lines tangent to 2n-2 spheres in Rn, with Thorsten Theobald, Trans. Amer. Math. Soc., 354 (2002), 4815-4829.   Some pictures and computer code.
22	From enumerative geometry to solving systems of polynomial equations with Macaulay 2, in Computations in Algebraic Geometry with Macaulay 2, edited by D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels. Algorithms and Computation in Mathematics 8, Springer-Verlag, 2001. pp. 101-129. Macaulay 2 source code for computations in the paper. (This requires the Macaulay 2 package realroots.m2.)
23	Some real and unreal enumerative geometry for flag manifolds, Michigan Math Journal, 48 (2000) pp. 573-592.
24	Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro, Experimental Mathematics, 9, Number 2, (2000), pp. 161-182. An archive of the computations and Maple scripts for some proofs.
25	Real rational curves in Grassmannians, J. Amer. Math. Soc. 13 (2000), 333-341.
26	The special Schubert calculus is real, Electronic Research Announcements of the AMS, 5, 1999, pp. 35-39.
27	Some remarks on real and complex output feedback, with Joachim Rosenthal, Sys. & Control Lett., 33 (1998), pp. 73-80. A non-trivial real system that cannot be controlled by real output feedback.
28	Enumerative geometry for real varieties, in Algebraic Geometry, Santa Cruz 1995, ed. by János Kollár, Proceedings and Symposia in Pure Mathematics, 61, No. 1, AMS 1997. pp. 435-447. An appendix contains further discussion of work of Ronga, Tognoli, and Vust.
29	Real enumerative geometry and effective algebraic equivalence, J. Pure and Appl. Algebra, 117 & 118 (1997), 601-615.
30	Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math. J., 87, (1997), 59-85. Published version of Ph.D. Thesis. The 8 page supplement.