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Information Center for Mathematical Science

국내수학자

Information Center for Mathematical Science

국외수학자

Bernd Sturmfels
이름 영어이름 Bernd Sturmfels Bernd Sturmfels bernd@math.berkeley.edu Dept. of Mathematics University of California Berkeley
 Tropical implicitization and mixed fiber polytopes. Software for algebraic geometry IMA Vol. Math. Appl. (2008), Vol 148, Pages Toric geometry of cuts and splits. Special volume in honor of Melvin Hochster Michigan Math. J. (2008), Vol 57, Pages 689-709 Semidefinite representation of the $k$-ellipse. Algorithms in algebraic geometry IMA Vol. Math. Appl. (2008), Vol 146, Pages Elimination theory for tropical varieties Math. Res. Lett. (2008), Vol 15, Pages 543-562 The hyperdeterminant and triangulations of the 4-cube Math. Comp. (2008), Vol 77, Pages 1653-1679 With contributions and an editors' epilogue Notices Amer. Math. Soc. (2008), Vol 55, Pages 467-473 Three counter-examples on semi-graphoids. Combin. Probab. Comput. (2008), Vol 17, Pages 239-257 Epistasis and shapes of fitness landscapes Statist. Sinica (2007), Vol 17, Pages 1317-1342 Affine buildings and tropical convexity Albanian J. Math. (2007), Vol 1, Pages 187-211 Toward the human genotope Bull. Math. Biol. (2007), Vol 69, Pages 2723-2735 Hyperdeterminantal relations among symmetric principal minors J. Algebra (2007), Vol 316, Pages 634-648 Conjunctive Bayesian networks Bernoulli (2007), Vol 13, Pages 893-909 The cyclohedron test for finding periodic genes in time course expression studies Stat. Appl. Genet. Mol. Biol. (2007), Vol 6, Pages Computing the integer programming gap Combinatorica (2007), Vol 27, Pages 367-382 Tropical discriminants J. Amer. Math. Soc. (2007), Vol 20, Pages 1111-1133 The Newton polytope of the implicit equation Mosc. Math. J. (2007), Vol 7, Pages 327-346,35 The mathematics of phylogenomics SIAM Rev. (2007), Vol 49, Pages 3-31 Algebraic factor analysis: tetrads, pentads and beyond Probab. Theory Related Fields (2007), Vol 138, Pages 463-493 Computing tropical varieties J. Symbolic Comput. (2007), Vol 42, Pages 54-73 Monomial dynamical systems over finite fields Complex Systems (2006), Vol 16, Pages 333-342 On the toric algebra of graphical models Ann. Statist. (2006), Vol 34, Pages 1463-1492 Evolution on distributive lattices J. Theoret. Biol. (2006), Vol 242, Pages 409-420 Combinatorial secant varieties Pure Appl. Math. Q. (2006), Vol 2, Pages 867-891 The maximum likelihood degree Amer. J. Math. (2006), Vol 128, Pages 671-697 Minimizing polynomials via sum of squares over the gradient ideal Math. Program. (2006), Vol 106, Pages 587-606 Resultants in genetic linkage analysis J. Symbolic Comput. (2006), Vol 41, Pages 125-137 Biology Algebraic statistics for computational biology (2005), Vol , Pages 125-159 Computation Algebraic statistics for computational biology (2005), Vol , Pages 43-84 Algebra Algebraic statistics for computational biology (2005), Vol , Pages 85-124 Statistics Algebraic statistics for computational biology (2005), Vol , Pages 3-42 Phylogenetic algebraic geometry. Projective varieties with unexpected properties Walter de Gruyter GmbH & Co. KG, Berlin (2005), Vol , Pages Matroid polytopes, nested sets and Bergman fans Port. Math. (N.S.) (2005), Vol 62, Pages 437-468 What is dots a Gröbner basis? Notices Amer. Math. Soc. (2005), Vol 52, Pages 1199-1200 Solving the likelihood equations Found. Comput. Math. (2005), Vol 5, Pages 389-407 On the rank of a tropical matrix. Combinatorial and computational geometry Math. Sci. Res. Inst. Publ. (2005), Vol 52, Pages Algebraic geometry of Bayesian networks J. Symbolic Comput. (2005), Vol 39, Pages 331-355 The graph of monomial ideals J. Pure Appl. Algebra (2005), Vol 201, Pages 250-263 First steps in tropical geometry. Idempotent mathematics and mathematical physics Contemp. Math. (2005), Vol 377, Pages Combinatorial commutative algebra Graduate Texts in Mathematics (2005), Vol 227, Pages Erratum for: ''Tropical convexity'' Doc. Math. (2004), Vol 9, Pages 205-206 Parametric inference for biological sequence analysis Proc. Natl. Acad. Sci. USA (2004), Vol 101, Pages 16138-1614 Tropical geometry of statistical models Proc. Natl. Acad. Sci. USA (2004), Vol 101, Pages 16132-1613 Classification of six-point metrics Electron. J. Combin. (2004), Vol 11, Pages Algebraic recipes for integer programming Proc. Sympos. Appl. Math. (2004), Vol 61, Pages Short rational functions for toric algebra and applications J. Symbolic Comput. (2004), Vol 38, Pages 959-973 Multigraded Hilbert schemes J. Algebraic Geom. (2004), Vol 13, Pages 725-769 Supernormal vector configurations. J. Algebraic Combin. (2004), Vol 19, Pages 297-313 Tropical convexity. Doc. Math. (2004), Vol 9, Pages 1-27 The tropical Grassmannian Adv. Geom (2004), Vol 4, Pages 389-411 Minimizing polynomial functions DIMACS Ser. Discrete Math. Theoret. Comput. Sci. (2003), Vol 60, Pages Algebraic unimodular counting. Algebraic and geometric methods in discrete optimization Math. Program. (2003), Vol 96, Pages 183-203 Higher Lawrence configurations J. Combin. Theory Ser. A (2003), Vol 103, Pages 151-164 Alexander duality in subdivisions of Lawrence polytopes Adv. Geom. (2003), Vol 3, Pages 177-189 Toric hyperKähler varieties Doc. Math. (2002), Vol 7, Pages 495-534 Algorithms for the toric Hilbert scheme. Computations in algebraic geometry with Macaulay Algorithms Comput. Math. (2002), Vol 8, Pages Teaching the geometry of schemes. Computations in algebraic geometry with Macaulay Algorithms Comput. Math. (2002), Vol 8, Pages Ideals, varieties and Macaulay 2. Computations in algebraic geometry with Macaulay Algorithms Comput. Math. (2002), Vol 8, Pages Elimination theory in codimension 2. J. Symbolic Comput. (2002), Vol 34, Pages 119-135 Solving systems of polynomial equations CBMS Regional Conference Series in Mathematics (2002), Vol 97, Pages Binomial residues Ann. Inst. Fourier (Grenoble) (2002), Vol 52, Pages 687-708 Syzygies of oriented matroids Duke Math. J. (2002), Vol 111, Pages 287-317 Gröbner bases of abelian matrix groups. Symbolic computation: solving equations in algebra, geometry, and engineering Contemp. Math. (2001), Vol 286, Pages Rational hypergeometric functions Compositio Math. (2001), Vol 128, Pages 217-239 Syzygies of unimodular Lawrence ideals J. Reine Angew. Math. (2001), Vol 534, Pages 169-186 A sagbi basis for the quantum Grassmannian J. Pure Appl. Algebra (2001), Vol 158, Pages 347-366 Four counterexamples in combinatorial algebraic geometry J. Algebra (2000), Vol 230, Pages 2802-294 Generic and cogeneric monomial ideals. Symbolic computation in algebra, analysis, and geometry J. Symbolic Comput. (2000), Vol 29, Pages 691-708 Solving algebraic equations in terms of $scr A$-hypergeometric series. Formal power series and algebraic combinatorics Discrete Math. (2000), Vol 210, Pages 171-181 Gröbner deformations of hypergeometric differential equations Algorithms and Computation in Mathematics (2000), Vol 6, Pages Monomial ideals and planar graphs. Applied algebra, algebraic algorithms and error-correcting codes Lecture Notes in Comput. Sci. (1999), Vol 1719, Pages Oriented matroids Encyclopedia of Mathematics and its Applications (1999), Vol 46, Pages The co-Scarf resolution. Commutative algebra, algebraic geometry, and computational methods Springer, Singapore (1999), Vol , Pages Cutting corners Adv. in Appl. Math. (1999), Vol 23, Pages 29-48 Hypergeometric polynomials and integer programming Compositio Math. (1999), Vol 115, Pages 185-204 Gröbner bases and hypergeometric functions. Gröbner bases and applications London Math. Soc. Lecture Note Ser. (1998), Vol 251, Pages Numerical Schubert calculus. Symbolic numeric algebra for polynomials J. Symbolic Comput. (1998), Vol 26, Pages 767-788 deformations of regular holonomic systems Proc. Japan Acad. Ser. A Math. Sci. (1998), Vol 74, Pages 111-113 Polynomial equations and convex polytopes Amer. Math. Monthly (1998), Vol 105, Pages 907-922 Cellular resolutions of monomial modules J. Reine Angew. Math. (1998), Vol 502, Pages 123-140 Syzygies of codimension $2$ lattice ideals Math. Z. (1998), Vol 229, Pages 163-194 A toric ring with irrational Poincaré-Betti series C. R. Acad. Sci. Paris Sér. I Math. (1998), Vol 326, Pages 141-146 Lattice walks and primary decomposition. Mathematical essays in honor of Gian-Carlo Rota Progr. Math. (1998), Vol 161, Pages Monomial resolutions Math. Res. Lett. (1998), Vol 5, Pages 31-46 Residues and esultants J. Math. Sci. Univ. Tokyo (1998), Vol 5, Pages 119-148 Algebraic algorithms for sampling from conditional distributions Ann. Statist. (1998), Vol 26, Pages 363-397 Introduction to resultants. Applications of computational algebraic geometry Proc. Sympos. Appl. Math. (1998), Vol 53, Pages How to shell a monoid Math. Ann. (1998), Vol 310, Pages 379-393 Generic lattice ideals J. Amer. Math. Soc. (1998), Vol 11, Pages 363-373 Non-commutative Gröbner bases for commutative algebras Proc. Amer. Math. Soc. (1998), Vol 126, Pages 687-691 Hypergeometric polynomials and integer programming. (Japanese) Research on the theory and applications of computer algebra Sūrikaisekikenkyūsho Kōkyūroku (1997), Vol , Pages 31-33 Equations defining toric varieties. Algebraic geometry Proc. Sympos. Pure Math., (1997), Vol 62, Pages Structural Gröbner basis detection Appl. Algebra Engrg. Comm. Comput. (1997), Vol 8, Pages 257-263 Variation of cost functions in integer programming Math. Programming (1997), Vol 77, Pages 357-387 Computing Hopf bifurcations. I. SIAM J. Numer. Anal. (1997), Vol 34, Pages 1-21 Bernstein's theorem in affine space Discrete Comput. Geom. (1997), Vol 17, Pages 137-141 Mixed monomial bases. Algorithms in algebraic geometry and applications Progr. Math. (1996), Vol 143, Pages Intersection theory on toric varieties Topology (1997), Vol 36, Pages 335-353 Computing multidimensional residues. Algorithms in algebraic geometry and applications Progr. Math. (1996), Vol 143, Pages A normal form algorithm for modules over $k[x,y]/langle xyrangle$ J. Algebra (1996), Vol 184, Pages 1001-1024 Primitive partition identities. Bolyai Soc. Math. Stud., 2 (1996.), Vol , Pages Binomial ideals Duke Math. J. (1996), Vol 84, Pages 1-45 Gröbner bases and convex polytopes University Lecture Series (1996), Vol 8, Pages The polytope of all triangulations of a point configuration Doc. Math. (1996), Vol 1, Pages 103-119 Introduction to Chow forms. Invariant methods in discrete and computational geometry Kluwer Acad. Publ., Dordrecht (1995), Vol , Pages GRIN: an implementation of Gröbner bases for integer programming. Integer programming and combinatorial optimization Lecture Notes in Comput. Sci. (1995), Vol 920, Pages Initial complexes of prime ideals Adv. Math. (1995), Vol 116, Pages 365-376 Gröbner bases of lattices, corner polyhedra, and integer programming Beiträge Algebra Geom. (1995), Vol 36, Pages 281-298 On vector partition functions J. Combin. Theory Ser. A (1995), Vol 72, Pages 302-309 Gröbner bases and triangulations of the second hypersimplex. Combinatorica (1995), Vol 15, Pages 409-424 Bounds on degrees of projective schemes Math. Ann. (1995), Vol 302, Pages 417-432 Intersection theory on spherical varieties J. Algebraic Geom. (1995), Vol 4, Pages 181-193 A polyhedral method for solving sparse polynomial systems Math. Comp. (1995), Vol 64, Pages 1541-1555 Iterated fiber polytopes Mathematika (1994), Vol 41, Pages 348-363 Viro's theorem for complete intersections Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1994), Vol 21, Pages 377-386 On the number of real roots of a sparse polynomial system Fields Inst. Commun. (1994), Vol 3, Pages A note on lattice simplices and toric varieties Amer. J. Math. (1994), Vol 116, Pages 1337-1339 Minimal polynomials and sparse resultants. Zero-dimensional schemes (Ravello, 1992), (1994), Vol , Pages 317-324 A quantitative Steinitz' theorem. Festschrift on the occasion of the 65th birthday of Otto Krötenheerdt Beiträge Algebra Geom. (1994), Vol 35, Pages 125-129 Finding sparse systems of parameters J. Pure Appl. Algebra (1994), Vol 94, Pages 143-157 On the Newton polytope of the resultant J. Algebraic Combin. (1994), Vol 3, Pages 207-236 Multigraded resultants of Sylvester type J. Algebra (1994), Vol 163, Pages 115-127 Cellular strings on polytopes Proc. Amer. Math. Soc. (1994), Vol 122, Pages 549-555 A note on polynomial reduction. J. Symbolic Comput. (1993), Vol 16, Pages 273-277 Algorithms in invariant theory Texts and Monographs in Symbolic Computation. (1993), Vol , Pages Sparse elimination theory. Computational algebraic geometry and commutative algebra Sympos Math., XXXIV (1993), Vol , Pages Product formulas for resultants and Chow forms Math. Z. (1993), Vol 214, Pages 377-396 Oriented matroids Encyclopedia of Mathematics and its Applications (1993), Vol 46, Pages A nonextremal Camion basis Linear Algebra Appl. (1993), Vol 187, Pages 195-199 Extension spaces of oriented matroids Discrete Comput. Geom. (1993), Vol 10, Pages 23-45 Minkowski addition of polytopes: computational complexity and applications to Gröbner bases SIAM J. Discrete Math. (1993), Vol 6, Pages 246-269 Maximal minors and their leading terms Adv. Math. (1993), Vol 98, Pages 65-112 Duality and minors of secondary polyhedra J. Combin. Theory Ser. B (1993), Vol 57, Pages 258-268 Asymptotic analysis of toric ideals. Mem. Fac. Sci. Kyushu Univ. Ser. A (1992), Vol 46, Pages 217-228 Chow polytopes and general resultants Duke Math. J. (1992), Vol 67, Pages 189-218 Algorithms for the Quillen-Suslin theorem J. Algebra (1992), Vol 145, Pages 231-239 Fiber polytopes Ann. of Math. (2) (1992), Vol 135, Pages 527-549 Unimodular fans, linear codes, and toric manifolds. Discrete and computational geometry DIMACS Ser. Discrete Math. Theoret. Comput. Sci. (1990), Vol 6, Pages 179--186 Computational algebraic geometry of projective configurations. Invariant-theoretic algorithms in geometry J. Symbolic Comput. (1991), Vol 11, Pages 595-618 Computing combinatorial decompositions of rings Combinatorica (1991), Vol 11, Pages 275-293 On the synthetic factorization of projectively invariant polynomials. Invariant-theoretic algorithms in geometry J. Symbolic Comput (1991), Vol 11, Pages 439-453 Stanley decompositions of the bracket ring Math. Scand. (1990), Vol 67, Pages 183-189 Smooth toric varieties with small Picard number are projective Topology (1991), Vol 30, Pages 289-299 On the topology and geometric construction of oriented matroids and convex polytopes Trans. Amer. Math. Soc. (1991), Vol 325, Pages 389-412 Constructions and complexity of secondary polytopes Adv. Math. (1990), Vol 83, Pages 155-179 Gröbner bases and Stanley decompositions of determinantal rings Math. Z. (1990), Vol 205, Pages 137-144 The intrinsic spread of a configuration in $Rsp d$ J. Amer. Math. Soc. (1990), Vol 3, Pages 639-651 All $11sb 3$ and $12sb 3$-configurations are rational Aequationes Math. (1990), Vol 39, Pages 254-260 Lawrence polytopes Canad. J. Math. (1990), Vol 42, Pages 62-79 Nonrealizability proofs in computational geometry Discrete Comput. Geom. (1990), Vol 5, Pages 333-350 Gröbner bases and invariant theory Adv. Math. (1989), Vol 76, Pages 245-259 Introduction to invariant theory in superalgebras. Invariant theory and tableaux IMA Vol. Math. Appl. (1990), Vol 19, Pages Computational synthetic geometry Lecture Notes in Mathematics (1989), Vol 1355, Pages On the matroid stratification of Grassmann varieties, specialization of coordinates, and a problem of N. White Adv. Math. (1989), Vol 75, Pages 202-211 Computing final polynomials and final syzygies using Buchberger's Gröbner bases method. Results Math. (1989), Vol 15, Pages 351-360 An infinite family of minorminimal nonrealizable $3$-chirotopes Math. Z. (1989), Vol 200, Pages 583-589 Arrangements of lines and pseudolines without adjacent triangles. J. Combin. Theory Ser. A (1989), Vol 50, Pages 24-32 Uniform oriented matroids without the isotopy property Discrete Comput. Geom. (1989), Vol 4, Pages 97-100 Gröbner bases of toric varieties Tohoku Math. J. (2) (1991), Vol 43, Pages 249-261 Quotients of toric varieties Math. Ann. (1991), Vol 290, Pages 643-655 Neighborly polytopes and oriented matroids. European J. Combin. (1988), Vol 9, Pages 537-546 Tridiagonalization of complex matrices and a problem of Longstaff Linear Algebra Appl. (1988), Vol 109, Pages 165-166 Totally positive matrices and cyclic polytopes. Proceedings of the Victoria Conference on Combinatorial Matrix Analysis Linear Algebra Appl. (1988), Vol 107, Pages 275-281 Simplicial cells in arrangements and mutations of oriented matroids. Geom. Dedicata (1988), Vol 27, Pages 153-170 Some applications of affine Gale diagrams to polytopes with few vertices SIAM J. Discrete Math. (1988), Vol 1, Pages 121-133 Aspects of computational synthetic geometry. I. Algorithmic coordinatization of matroids. Computer aided geometric reasoning (1987), Vol I, II, Pages 57--86 Cyclic polytopes and $d$-order curves Geom. Dedicata (1987), Vol 24, Pages 103--107 Polytopal and nonpolytopal spheres: an algorithmic approach Israel J. Math. (1987), Vol 57, Pages 257--271 On the decidability of Diophantine problems in combinatorial geometry Bull. Amer. Math. Soc. (N.S.) (1987), Vol 17, Pages 121--124 Boundary complexes of convex polytopes cannot be characterized locally J. London Math. Soc. (2) (1987), Vol 35, Pages 314--326 Central and parallel projections of polytopes Discrete Math. (1986), Vol 62, Pages 315--318 On the coordinatization of oriented matroids. Discrete Comput. Geom. (1986), Vol 1, Pages 293--306 Reell realisierbare orientierte Matroide. Bayreuth. Math. Schr. (1986), Vol 21, Pages 1-13 The graph of monomial ideals (), Vol , Pages Cellular Resolutions of Monomial Modules J. reine angew. Math. (1998), Vol 502, Pages 123-140 Generic lattice ideals J. Amer. Math. Soc. (1998), Vol 11, Pages 363-373 Syzygies of codimension 2 lattice ideals Math. Z. (1998), Vol 229, Pages 163-194 Algorithms in invariant theory. Texts Monogr. Symbol. Comput. (2008), Vol , Pages vi+197 Normal toric ideals of low codimension. J. Pure Appl. Algebra (2009), Vol 213, Pages 1636-1641 Open problems in algebraic statistics. IMA Vol. Math. Appl. (2009), Vol , Pages 351-363 Tropical mathematics. Math. Mag. (2009), Vol 82, Pages 163-173 Guest editors' foreword [Special issue dedicated to the memory of Victor Klee]. Discrete Comput. Geom. (2009), Vol 42, Pages 131 How to draw tropical planes. Electron. J. Combin. (2009), Vol 16, Pages 26 Matrix cubes parameterized by eigenvalues. SIAM J. Matrix Anal. Appl. (2009), Vol 31, Pages 755-766 Convex rank tests and semigraphoids. SIAM J. Discrete Math. (2009), Vol 23, Pages 1117-1134 Toric dynamical systems. J. Symbolic Comput. (2009), Vol 44, Pages 1551-1565 Marginal likelihood integrals for mixtures of independence models. J. Mach. Learn. Res. (2009), Vol 10, Pages 1611–1631 Polynomial relations among principal minors of a $4\times 4$-matrix. J. Algebra (2009), Vol 322, Pages 4121-4131 Computer algebra in systems biology. Amer. Math. Monthly (2009), Vol 116, Pages 882-891 Lectures on algebraic statistics. Oberwolfach Semin. (2009), Vol , Pages The algebraic degree of semidefinite programming. Math. Program. (2010), Vol 122, Pages 379–405 Commuting birth-and-death processes. Ann. Appl. Probab. (2010), Vol 20, Pages 238-266 Sagbi bases of Cox-Nagata rings. J. Eur. Math. Soc. (JEMS) (2010), Vol 12, Pages 429-459 The Hilbert scheme of the diagonal in a product of projective spaces. Int. Math. Res. Not. IMRN (2010), Vol , Pages 1741–1771 Blow-ups of $\Bbb P^{n-3}$ at $n$ points and spinor varieties. J. Commut. Algebra (2010), Vol 2, Pages 223-244 Siphons in chemical reaction networks. Bull. Math. Biol. (2010), Vol 72, Pages 1448-1463 Geometry of the restricted Boltzmann machine. Contemp. Math. (2010), Vol , Pages 135–153 Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry. Ann. Inst. Statist. Math. (2010), Vol 62, Pages 603-638 Dualities in convex algebraic geometry Rend. Mat. Appl. (7) (2010), Vol 30, Pages 285-327 Quartic curves and their bitangents. J. Symbolic Comput. (2011), Vol 46, Pages 712-733 Tropical quadrics through three points. Linear Algebra Appl. (2011), Vol 435, Pages 1778-1785 Orbitopes. Mathematika (2011), Vol 57, Pages 275-314 The universal Kummer threefold. Exp. Math. (2013), Vol 22, Pages 327–362 Elliptic curves in honeycomb form. Algebraic and combinatorial aspects of tropical geometry, 87–107 Contemp. Math. (2013), Vol , Pages Dualities. Semidefinite optimization and convex algebraic geometry, 203–249 MOS-SIAM Ser. Optim. (2013), Vol , Pages A Hilbert scheme in computer vision. Canad. J. Math. (2013), Vol 65, Pages 961–988 Mixed discriminants. Math. Z. (2013), Vol 274, Pages 761–778 The entropic discriminant. Adv. Math. (2013), Vol 244, Pages 678–707 Capacity pre-log of noncoherent SIMO channels via Hironaka's theorem. IEEE Trans. Inform. Theory (2013), Vol 59, Pages 4213–4229 Monomials, binomials and Riemann-Roch. J. Algebraic Combin. (2013), Vol 37, Pages 737–756 Combinatorial types of tropical eigenvectors. Bull. Lond. Math. Soc. (2013), Vol 45, Pages 27–36 Toric cubes. Rend. Circ. Mat. Palermo (2) (2013), Vol 62, Pages 67–78 Binary cumulant varieties. Ann. Comb. (2013), Vol 17, Pages 229–250 Matrices with eigenvectors in a given subspace. Proc. Amer. Math. Soc. (2013), Vol 141, Pages 1219–1232 The number of eigenvalues of a tensor. Linear Algebra Appl. (2013), Vol 438, Pages 942–952 Geometry of higher-order Markov chains. J. Algebr. Stat. (2012), Vol 3, Pages 1–10 Algebraic boundaries of Hilbert's SOS cones. Compos. Math. (2012), Vol 148, Pages 1717–1735 Graphical models for correlated defaults. Math. Finance (2012), Vol 22, Pages 621–644 The central curve in linear programming. Found. Comput. Math. (2012), Vol 14, Pages 509–540 Computing linear matrix representations of Helton-Vinnikov curves. Mathematical methods in systems, optimization, and control, 259–277 Oper. Theory Adv. Appl. (2012), Vol , Pages Commutative algebra of statistical ranking. J. Algebra (2012), Vol 361, Pages 264–286 On the convex hull of a space curve. Adv. Geom. (2012), Vol 12, Pages 157–178 Quartic spectrahedra. Math. Program. (2015), Vol 151, Pages 585–612 Tensors of nonnegative rank two. Linear Algebra Appl. (2015), Vol 473, Pages 37–53 Introduction to tropical geometry. Graduate Studies in Mathematics (2015), Vol 161, Pages Moduli of tropical plane curves. Res. Math. Sci. (2015), Vol 2, Pages Art. 4, 31 Fixed points EM algorithm and nonnegative rank boundaries. Ann. Statist. (2015), Vol 43, Pages 422–461 Exact solutions in structured low-rank approximation. SIAM J. Matrix Anal. Appl. (2014), Vol 35, Pages 1521–1542 Maximum likelihood for matrices with rank constraints. J J. Algebr. Stat. (2014), Vol 5, Pages 18–38 Hypersurfaces and their singularities in partial correlation testing. Found. Comput. Math. (2014), Vol 14, Pages 1079–1116 Tropicalization of classical moduli spaces. Math. Comput. Sci. (2014), Vol 8, Pages 119–145