Frank Sottile: Papers in Geometry

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

1	Khovanskii-Rolle continuation for real solutions, with Daniel J. Bates, Foundations of Computational Mathematics. DOI: 10.1007/s10208-011-9097-1. 25 pages.
2	Experimentation at the Frontiers of Reality in Schubert Calculus with Chris Hillar, Luis García-Puente, Abraham Martín del Campo-Sanchez, James Ruffo, Zach Teitler, and Stephen L. Johnson. Contemporary Mathmematics, 317, Amer. Math. Soc., Providence, RI, 2010, pp. 365-380.
3	Fewnomial bounds for completely mixed polynomial systems, with Frédéric Bihan, Advances in Geometry, (2011), DOI:10.1515/ADVGEOM.2011.019.
4	Frontiers of Reality in Schubert Calculus Bulletin of the AMS, 47, No 1. (2010), 31-71.
5	Some geometrical aspects of control points for toric patches with Gheorghe Craciun and Luis Garcia, in Mathematical Methods for Curves and Surfaces, Lecture Notes in Computer Science 5862, Springer 2010, 111–135.
6	Betti number bounds for fewnomial hypersurfaces via stratified Morse theory, with Frédéric Bihan, Proc. Amer. Math. Soc., 137, No. 9 (2009), 2825-2833.
7	General isotropic flags are general (for Grassmannian Schubert calculus), Journal of Algebraic Geometry, 19 (2010), 367-370.
8	Linear precision for toric surface patches, with Kristian Ranestad and Hans-Christian Graf von Bothmer. 25 pages. ArXiv.org/0806.3230. Foundations of Computational Mathematics, 10, Issue 1 (2010), 37--66, DOI: 10.1007/s10208-009-9052-6.
9	Linear precision for parametric patches with Luis Garcia Advances in Computational Mathematics. 33 (2010), Page 191--214. DOI: 10.1007/s10444-009-9126-7
10	Galois groups of Schubert problems via homotopy computation, with Anton Leykin, Mathematics of Computation, 78 (2009) 1749--1765.
11	A Littlewood-Richardson rule for Grassmannian permutations with Kevin Purbhoo. Proc. Amer. Math. Soc., 137 (2009) 1875--1882.
12	The equivariant cohomology rings of quot schemes, with Tom Braden, and Linda Chen. Pacific Journal of Mathematics, 238, No. 2 (2008) 201--232.
13	Gale duality for complete intersections, with Frédéric Bihan, Annales de l'Institut Fourier, Tome 58 (2008) fasicule 3, 877--891. A Singular script which computes examples of the Kouchnirenko Theorem for systems of Master functions.
14	Convex Hulls of Orbits and Orientations of a Moving Protein Domain, with Marco Longinetti and Luca Sgheri, Discrete and Computational Geometry, 43 No. 1 (2010), 54–77.
15	The recursive nature of cominuscule Schubert calculus with Kevin Purbhoo Advances in Mathematics, 217 (2008), pp. 1962--2004.
16	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.
17	Real Hessian Curves with Adriana Ortiz-Rodríguez. Boletín de la Sociedad Mathemática Mexicana, Volume 13 Number 3 (2007), 157--166. Companion web page.
18	Bounds on the number of real solutions to polynomial equations, with Daniel J. Bates, Frédéric Bihan, IMRN, 2007, 2007:rnm114-7.
19	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. 15--20, Springer, New York, 2007.
20	New fewnomial upper bounds from Gale dual polynomial systems, with Frédéric Bihan, Moscow Mathematics Journal, 7 (2007), Number 3, 387--407.
21	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.
22	A Pieri-type formula for the K-theory of a flag manifold, with Cristian Lenart, Trans. Amer. Math. Soc., Trans. Amer. Math. Soc. 359 (2007), 2317--2342.
23	Grothendieck Polynomials via Permutation Patterns and chains in the Bruhat Order, with Cristian Lenart and Shawn Robinson. American Journal of Mathematics, 128, No. 4, (2006), 805--848.
24	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.
25	Cremona Convexity, Frame Convexity, and a Theorem of Santaló, with Jacob E. Goodman, Andreas Holmsen, Richard Pollack, and Kristian Ranestad. Advances in Geometry, 6, No. 3, (2006), 301--322.
26	Polynomial systems with few real zeroes, with Benoît Bertrand and Frédéric Bihan, Mathematisches Zeitschrift, 253 (2006), no. 2, 361--385.
27	Lower Bounds for Real Solutions to Sparse Polynomial Systems, with Evgenia Soprunova, Advances in Mathematics, Volume 204, Issue 1, 1 August 2006, 116--151.
28	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.
29	Quiver Coefficients are Schubert Structure Constants, with Anders Buch and Alex Yong, Mathematics Research Letters, Volume 12, Issue 4, (2005) 567--574.
30	Real k-flats tangent to quadrics in Rn, with Thorsten Theobald. Proc. Amer. Math. Soc., 133 (2005), 2835--2844.
31	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.
32	Maximally inflected real rational curves, with Viatcheslav Kharlamov. Moscow Mathematics Journal 3 (2003), no. 3, 947--987, 1199--1200.     Companion Web Page.
33	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.)
34	Elementary transversality in the Schubert calculus in any characteristic. Michigan Math Journal, 51 (2003), 651--666.
35	Common transversals and tangents to two lines and two quadrics in P3, with Gábor Megyesi and Thorsten Theobald. Discrete and Computational Geometry, 30, (2003), 543–571.     Companion Web Page.
36	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.
37	Lines tangent to 2n-2 spheres in Rn, with Thorsten Theobald, Trans. Amer. Math. Soc., 354 (2002), 4815--4829.   Some pictures and computer code.
38	A Pieri-type formula for isotropic flag manifolds, with Nantel Bergeron, Trans. Amer. Math. Soc., 354 No. 7, (2002), 2659--2705.
39	Skew Schubert functions and the Pieri formula for flag manifolds, with Nantel Bergeron. Trans. Amer. Math. Soc., 354 No. 2, (2002), 651--673.
40	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.)
41	Rational curves on Grassmannians: systems theory, reality, and transversality, In "Advances in Algebraic Geometry Motivated by Physics", ed. by Emma Previato, Contemporary Mathematics, 276, 2001, pp. 9--42.
42	A sagbi basis for the quantum Grassmannian, with Bernd Sturmfels. J. Pure and Appl. Algebra, 158, 24 April 2001 pp. 347-366.
43	Some real and unreal enumerative geometry for flag manifolds, Michigan Math Journal, 48 (2000) pp. 573-592.
44	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.
45	Real rational curves in Grassmannians, J. Amer. Math. Soc. 13 (2000), 333-341.
46	Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloquium Mathematicum, 82 (1999), pp. 49--63.
47	The special Schubert calculus is real, Electronic Research Announcements of the AMS, 5, 1999, pp. 35-39.
48	Numerical Schubert calculus, with Birkett Huber and Bernd Sturmfels, Journal of Symbolic Computation, 26, (1998) pp. 767-788.
49	Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, with Nantel Bergeron, Duke Math. J., 95, (1998), pp. 373-423. A 16 page supplement. A description of how Table 1 was generated.
50	Pieri's formula via explicit rational equivalence, Canad. J. Math., 46 (1997), pp. 1281-1298. A 5 page supplement.
51	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.
52	Real enumerative geometry and effective algebraic equivalence, J. Pure and Appl. Algebra, 117 & 118 (1997), 601-615.
53	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.
54	Pieri's formula for flag manifolds and Schubert polynomials, Annales de l'Institut Fourier, 46 (1996), pp. 89--110.
55	Intersection theory on spherical varieties, with Wm. Fulton, R. MacPherson, and Bernd Sturmfels, J. Alg. Geom., 4, (1995), pp. 181-193.