Frank Sottile: Publications and Preprints

1	Real Solutions to Equations From Geometry, University Lecture Series volume 57, AMS, 2011. 200+viii pages. Older version: math.AG/0609829.

2	Discriminant coamoebas through homology with Mikael Passare†
3	Static skew-symmetric output feedback control with Chris Hillar 16 pages. 2011, arXiv.org/1111.3501
4	Non-archimedean coamoebae, with Mounir Nisse, 2011. arxiv.org/1110.1033.
5	Injectivity of 2D Toric Bézier patches with Chungang Zhu, 4 pages. CAD/Graphics 2011, presented. ArXiv.org/1106.2877.
6	The phase limit set of a variety, with Mounir Nisse, 2011. arxiv.org/1106.0096.
7	Orientability of real toric varieties, with Jenya Soprunova, 2011. arxiv.org/1102.3045.
8	Arithmetic Toric Varieties with E. Javier Elizondo, Paulo Lima-Filho, and Zach Teitler. 27 pages. arXiv:1003.5141.

9	The Secant Conjecture in the real Schubert calculus with Luis García-Puente, Nickolas Hein, Chris Hillar, Abraham Martín del Campo-Sanchez, James Ruffo, and Zach Teitler. Experimental Mathematics, to appear, 19 pages. ArXiv.org/1010.0665.
10	Cofree compositions of Coalgebras with Stefan Forcey and Aaron Lauve. Annals of Combinatorics, to appear. arXiv.org/1012.3483.
11	alphaCertified: certifying solutions to polynomial systems with Jonathan Hauenstein. ACM Transactions on Matematical Software, to appear. 21 pages, 2011. arXiv.org/1011.1091. Software and guide.
12	Toric degenerations of Bézier patches with Chungang Zhu, and Luis Garcia. 11 pages. ACM Transactions on Graphics, to appear. ArXiv.org/1006.4903.

13	Dense Fewnomials with Korben Rusek and Jeanette Shakalli. Randomization, Relaxation, and Complexity, Leonid Gurvits, Philippe Pébay, J. Maurice Rojas, and David Thompson, eds., Contemporary Mathematics 556, AMS, 2011, pp. 167–186.
14	Khovanskii-Rolle continuation for real solutions, with Daniel J. Bates, Foundations of Computational Mathematics. DOI: 10.1007/s10208-011-9097-1. 25 pages.
15	Orbitopes with Raman Sanyal and Bernd Sturmfels. Mathematika, 57 (2011), issue 02, 275–314.
16	Fewnomial bounds for completely mixed polynomial systems, with Frédéric Bihan, Advances in Geometry, (2011), DOI:10.1515/ADVGEOM.2011.019.
17	Hopf Structures on the Multiplihedra with Stefan Forcey and Aaron Lauve. SIAM Journal on Discrete Mathematics, 24, No. 4, (2010), 1250-1271. Published online, DOI: 10.1137/090776834.
18	Solving Schubert Problems with Littlewood-Richardson Homotopies, with Ravi Vakil and Jan Verschelde. Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, edited by Stephen M. Watt, pages 179-186, ACM 2010.
19	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.
20	A Skew Littlewood-Richardson rule from Hopf algebras with Thomas Lam and Aaron Lauve. IMRN, doi:10.1093/imrn/rnq104, Published June 2, 2010.
21	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.
22	General isotropic flags are general (for Grassmannian Schubert calculus), Journal of Algebraic Geometry, 19 (2010), 367-370.
23	Frontiers of Reality in Schubert Calculus Bulletin of the AMS, 47, No 1. (2010), 31-71.
24	Betti number bounds for fewnomial hypersurfaces via stratified Morse theory, with Frédéric Bihan, Proc. Amer. Math. Soc., 137, No. 9 (2009), 2825-2833.
25	Linear precision for parametric patches with Luis Garcia Advances in Computational Mathematics. 33 (2010), Page 191--214. DOI: 10.1007/s10444-009-9126-7
26	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.
27	Galois groups of Schubert problems via homotopy computation, with Anton Leykin, Mathematics of Computation, 78 (2009) 1749--1765.
28	A Littlewood-Richardson rule for Grassmannian permutations with Kevin Purbhoo. Proc. Amer. Math. Soc., 137 (2009) 1875--1882.
29	(Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones), with Matthias Beck and Christian Haase. Mathematical Intelligencer, 31 (2009), No. 1, 9--17.
30	The equivariant cohomology rings of quot schemes, with Tom Braden, and Linda Chen. Pacific Journal of Mathematics, 238, No. 2 (2008) 201--232.
31	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.
32	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.
33	The recursive nature of cominuscule Schubert calculus with Kevin Purbhoo Advances in Mathematics, 217 (2008), pp. 1962--2004.
34	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.
35	Real Hessian Curves with Adriana Ortiz-Rodríguez. Boletín de la Sociedad Mathemática Mexicana, Volume 13 Number 3 (2007), 157--166. Companion web page.
36	Bounds on the number of real solutions to polynomial equations, with Daniel J. Bates, Frédéric Bihan, IMRN, 2007, 2007:rnm114-7.
37	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.
38	New fewnomial upper bounds from Gale dual polynomial systems, with Frédéric Bihan, Moscow Mathematics Journal, 7 (2007), Number 3, 387--407.
39	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.
40	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.
41	Irrational proofs of three theorems of Stanley, with Matthias Beck, European Journal of Combinatorics, 28, Issue 1, January 2007, 403--409.
42	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.
43	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.
44	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.
45	Polynomial systems with few real zeroes, with Benoît Bertrand and Frédéric Bihan, Mathematisches Zeitschrift, 253 (2006), no. 2, 361--385.
46	Lower Bounds for Real Solutions to Sparse Polynomial Systems, with Evgenia Soprunova, Advances in Mathematics, Volume 204, Issue 1, 1 August 2006, 116--151.
47	Cocommutative Hopf algebras of permutations and trees, with Marcelo Aguiar. Journal of Algebraic Combinatorics, Volume 22 (2005), Number 4, pp. 451--470.
48	Combinatorial Hopf Algebras and generalized Dehn-Sommerville relations, with Marcelo Aguiar and Nantel Bergeron, Compositio Mathematica, 142, January 2006, 1--30.
49	Structure of the Loday-Ronco Hopf algebra of trees, with Marcelo Aguiar, Journal of Algebra, Volume 295, Issue 2 , 15 January 2006, 473--511.
50	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.
51	Quiver Coefficients are Schubert Structure Constants, with Anders Buch and Alex Yong, Mathematics Research Letters, Volume 12, Issue 4, (2005) 567--574.
52	Real k-flats tangent to quadrics in Rn, with Thorsten Theobald. Proc. Amer. Math. Soc., 133 (2005), 2835--2844.
53	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.
54	Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, with Marcelo Aguiar, Advances in Mathematics 191, (2005), 225--275.
55	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.
56	Maximally inflected real rational curves, with Viatcheslav Kharlamov. Moscow Mathematics Journal 3 (2003), no. 3, 947--987, 1199--1200.     Companion Web Page.
57	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.)
58	Elementary transversality in the Schubert calculus in any characteristic. Michigan Math Journal, 51 (2003), 651--666.
59	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.
60	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.
61	Skew Schubert polynomials, with Cristian Lenart, Proc. Amer. Math. Soc., 131 (2003), 3319-3328.
62	Lines tangent to 2n-2 spheres in Rn, with Thorsten Theobald, Trans. Amer. Math. Soc., 354 (2002), 4815--4829.   Some pictures and computer code.
63	A Pieri-type formula for isotropic flag manifolds, with Nantel Bergeron, Trans. Amer. Math. Soc., 354 No. 7, (2002), 2659--2705.
64	Shifted quasi-symmetric functions and the Hopf algebra of peak functions with Nantel Bergeron, S. Mykytiuk, and S. van Willigenburg, Discrete Math., 256 (2002), 57--66.
65	Skew Schubert functions and the Pieri formula for flag manifolds, with Nantel Bergeron. Trans. Amer. Math. Soc., 354 No. 2, (2002), 651--673.
66	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.)
67	Complementary Algorithms for Tableaux, with Tom Roby, Jeff Stroomer, and Julian West. J. Combin. Th. Ser. A, 96, No. 1, October 2001, 127-161.
68	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.
69	A sagbi basis for the quantum Grassmannian, with Bernd Sturmfels. J. Pure and Appl. Algebra, 158, 24 April 2001 pp. 347-366.
70	Some real and unreal enumerative geometry for flag manifolds, Michigan Math Journal, 48 (2000) pp. 573-592.
71	Non-commutative Pieri operators on posets, Nantel Bergeron, S. Mykytiuk, and S. van Willigenburg, J. Combin. Th. Ser. A, Vol. 91, No. 1/2, Aug 2000, pp. 84-110.
72	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.
73	Real rational curves in Grassmannians, J. Amer. Math. Soc. 13 (2000), 333-341.
74	Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloquium Mathematicum, 82 (1999), pp. 49--63.
75	Hopf algebras and edge-labeled posets, with Nantel Bergeron, Journal of Algebra, 216 (1999), pp. 641-651.
76	A monoid for the Grassmannian Bruhat order, with Nantel Bergeron, European Journal of Combinatorics, 20, (1999), pp. 197-211.
77	The special Schubert calculus is real, Electronic Research Announcements of the AMS, 5, 1999, pp. 35-39.
78	Numerical Schubert calculus, with Birkett Huber and Bernd Sturmfels, Journal of Symbolic Computation, 26, (1998) pp. 767-788.
79	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.
80	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.
81	Pieri's formula via explicit rational equivalence, Canad. J. Math., 46 (1997), pp. 1281-1298. A 5 page supplement.
82	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.
83	Real enumerative geometry and effective algebraic equivalence, J. Pure and Appl. Algebra, 117 & 118 (1997), 601-615.
84	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.
85	Tableau switching: algorithms and applications, with Georgia Benkart and Jeff Stroomer, J. Combin. Th. Ser. A, 76, (1996), pp. 11-43.
86	Pieri's formula for flag manifolds and Schubert polynomials, Annales de l'Institut Fourier, 46 (1996), pp. 89--110.
87	Intersection theory on spherical varieties, with Wm. Fulton, R. MacPherson, and Bernd Sturmfels, J. Alg. Geom., 4, (1995), pp. 181-193.
88	Real enumerative geometry for the Grassmannian of lines in projective space, Ph.D. Thesis, University of Chicago, August 1994. Published as Enumerative geometry for the real Grassmannian of lines in projective space

1	Injectivity of 2D Toric Bézier Patches, With Chungang Zhu, Extended abstract for CAD/Graphics, Jinan, China 2011.
2	Cofree compositions of coalgebras, With Stefan Forcey and Aaron Lauve. Extended abstract for FPSAC11, June 13--17, Reykjavik.
3	Toric Degenerations of Bézier Patches, With Luis García-Puente and Chungang Zhu. Extended abstract for MEGA 2011, Stockholm.
4	Solving Schubert Problems with Littlewood-Richardson Homotopies, With Ravi Vakil and Jan Verschelde. Extended abstract for ISSAC 10, Munich.
5	New Hopf Structures on Binary Trees, With Aaron Lauve and Stefan Forcey. Extended abstract for FPSAC09, July 2009, RISC, Hagenberg, Austria.
6	Gale duality for complete intersections, With Frédéric Bihan. Abstract for Oberwolfach report on the workshop on tropical geometry, 10--14 December 2007.
7	Computing monodromy via parallel homotopy continuation, With Anton Leykin. Extended abstract. For Parallel Symbolic Computation'07, 27 - 28 July 2007 University of Western Ontario May 2007.
8	The Horn recursion for Schur P- and Q- functions, With Kevin Purbhoo. Extended abstract. For the FPSAC`06 conference in San Diego, CA. February 2006.
9	Experimentation and Conjectures in the real Schubert calculus for flag manifolds, Extended abstract. Presented at the MEGA05 Conference. January 2005.
10	On the number of tangents to four triangles in three-dimensional space. With Olivier Devillers, Hervé Brönnimann, and Sylvain Lazard. Presented by Brönnimann at 16th Canadian Conference on Computational Geometry, Concordia University, Montreal. August 9-11, 2004.     Companion web page.
11	Transversals to line segments in R3. With Hervé Brönnimann, Hazel Everett, Sylvain Lazard, and Sue Whitesides. Presented by Brönnimann at 15th Canadian Conference on Computational Geometry, Dalhousie University, Halifax. 11-13 August, 2003.     Companion web page.
12	Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, with Marcelo Aguiar. Presented by Aguiar at Formal Power Series and Algebraic Combinatorics, Melbourne, Australia, 8-13 July 2002.
13	Jeux de Tableaux. With Tom Roby, Jeff Stroomer, and Julian West. Presented by Julian West at FPSAC00. Published in Proceedings, 12th International Conference, FPSAC`00, Moscow, June 2000, D. Krob, A.A. Mikhalev, A.V. Mikhalev, eds. Springer-Verlag, 2000. pp. 332-343.
14	Shifted quasi-symmetric functions and the Hopf algebra of peak functions, with Nantel Bergeron, S. Mykytiuk, and S. van Willigenburg, Presented by Mykytiuk at FPSAC, Barcelona, 7-11 June 1999.
15	Identities of Littlewood-Richardson coefficients for Schubert polynomials and orders on Sn , with Nantel Bergeron, Formal Power Series and Algebraic Combinatorics, 14-18 July, 1997, Universität Wien.
16	A geometric approach to the combinatorics of Schubert polynomials, Formal Power Series and Algebraic Combinatorics, May 29 - June 2, 1995, Marne-la-Valée, Paris.

1. Enumerative Real Algebraic Geometry. A web-based survey of the subject. Web version of paper by the same name. August 2002.
2. Some Maximally inflected curves. Work with Viatcheslav Kharlamov. This page accompanies the paper maximally Inflected Curves. June 2002.
3. Lines transversal to 2 lines and a quadric. Work with Gábor Megyesi and Thorsten Theobald. This page supplements the paper of the same name. April 2002.
4. The conjectures of Shapiro and Shapiro: an archive of computations. This describes the conjectures of Shapiro and Shapiro, providing more context than the paper Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro. Local version. April 2000.
5. Identities of Littlewood-Richardson Coefficients for Schubert polynomials. This describes how Nantel Bergeron and I generated Table 1 of Schubert polynomials, the Bruhat order, and the Geometry of Schubert varieties and discusses the Grassmannian Bruhat order. August 1998.
6. A non-trivial real system that cannot be controlled by real output feedback. An amplification of part of Some remarks on real and complex output feedback. June 1997.

1. Frontiers of Reality in Schubert Calculus. AMS Current Events Bulletin, January 2009.   .
2. Real Solutions to Equations from Geometry. Lecture notes from short course at the Centre Émile Borel, Institut Henri Poincaré, Given in November 2005.   .
3. Tropical Interpolation, Article for Autumn 2004 issue of the Emissary, the newsletter of MSRI. math.AG/0501146,   .
4. Letter to the Editor of the Notices of the AMS concerning the Digital Millenuim Coptright Act, Volume 43, Number 3, (2002), page 310. Text of letter. Also signed by George Avrunin, Anders Buch, and Andrew Hwang.
5. Supplement III, Encyclopaedia of Mathematics. M. Hazewinkel, Managing Editor Schubert Calculus 343-344, Schubert Cell 344-344, Schubert Cycle 344-345, and Schubert Polynomial 345-346. Kluwer Academic Publishers, 2002
6. Proceedings, 10 International Conference on Formal Power Series and Algebraic Combinatorics (10^ième conférence, Les Séries Formelles et la Combinatoire Algébriques) 14-19 June 1998, Fields Institute, Toronto. Ed. by Nantel Bergeron, Ian Goulden, and Frank Sottile. (See conference web pages:FPSAC'98, SFCA'98)
7. Starting Our Careers: A Collection of Essays and Advice on Professional Development from the Young Mathematicians' Network. Edited by Analissa Crandel and Curt Bennett. AMS 1999, 116pp.
   The following articles I wrote appear in that compilation:
   • Beating the Cost of Meetings Originally in CoYM v3.i1
   • Giving Math Talks, with Curt Bennett. Originally in CoYM v3.i4
   • AMS Special Sessions, with Curt Bennett. Originally in CoYM v3.i35
   • Getting Started in Research: a personal perspective Originally in CoYM v5.i6
8. A graduate student's view of the international congress of mathematicians, Mathematical Intelligencer, 9 No. 2 (1987) 33-35.