### Preprints

The slack realization space of a polytope,
(with João Gouveia, Antonio Macchia and Amy Wiebe), August 2017.

Symmetry in Turán sums of squares polynomials from flag algebras,
(with Annie Raymond and Mohit Singh), July 2015.

### Articles in print

** [1] ** An algebraic geometry
algorithm for scheduling in presence
of setups and correlated demands, (with Sridhar R. Tayur and
N.R.Natraj), *Mathematical Programming* **69**
(1995) 369-401.

**[2]** Groebner bases and
triangulations of the second hypersimplex,
(with Jesus de Loera and Bernd Sturmfels), *Combinatorica*
** 15 ** (1995) 409-424.

**[3]** A geometric Buchberger
algorithm for integer programming,
*Mathematics of Operations Research* ** 20 ** (1995)
864-884.

**[4]** Groebner bases in integer programming,
(with Serkan Hosten),
*OPTIMA Mathematical Programming Society Newsletter*
** 48 ** (1995).

**[5]** Test sets and inequalities for integer
programs, (with Robert Weismantel)
* Proceedings of the 5th International IPCO conference, Vancouver *,
LNCS **1084** (1996) 16-30.

**[6]** Variation of cost
functions in integer programming ,
(with Bernd Sturmfels), *Mathematical Programming*
** 77 ** (1997)
357-387.

**[7]** Truncated Groebner bases
for integer programming, (with Robert Weismantel), *Applicable
Algebra in Engineering, Communication and Computing* ** 8
** (1997) 241-257.

**[8]** Applications to integer programming,
in * Applications of Computational Algebraic Geometry, *
D.A. Cox and B. Sturmfels eds.,
AMS Proceedings of Symposia in Applied Mathematics ** 53 **
(1997) 119-142.

**[9]** Groebner bases in
integer programming ,
* Handbook of Combinatorial Optimization Vol. 1*,
D.-Z. Du and P.M.Pardalos eds., Kluwer Acad. Publ.,
Boston, MA (1998) 533-572.

**[10]** Groebner bases and integer programming,
(with Serkan Hosten), * Groebner Bases and
Applications*, B.Buchberger and F.Winkler eds.,
Cambridge University Press, London Math. Soc. Lecture Notes Series
** 251 ** (1998) 144-158.

**[11]** The associated primes
of initial ideals of lattice ideals , (with Serkan Hosten), *
Mathematical Research Letters * ** 6 ** (1999) 83-97.

**[12]** Standard pairs and group
relaxations in integer programming, (with Serkan Hosten), *
Journal of Pure and Applied Algebra * ** 139 **
(1999) 133-157.

**[13]** Computing Groebner
fans of toric ideals, (with Birk Huber),
* Experimental Mathematics * ** 9 ** (2000)
321-331.

**[14]** Algebraic methods in
integer programming , * Encyclopedia of
Optimization * (eds: C. Floudas and P. Pardalos), Kluwer Academic
Publishers, Dordrecht, 2001

**[15]** Combinatorics of the toric Hilbert
scheme , (with Diane Maclagan), * Discrete and Computational
Geometry * ** 27 ** (2002) 249-264.

**[16]** Algorithms for the toric Hilbert
scheme , (with Mike Stillman and Bernd Sturmfels), a chapter in
* Computations in Algebraic Geometry with Macaulay 2 * (eds:
D. Eisenbud, D. Grayson, M. Stillman and B. Sturmfels), Algorithms and
Computation in Mathematics Vol 8, Springer 2002. pp 179-213

**[17]**
The Hilbert zonotope and a polynomial
time algorithm for universal Groebner bases , (with
Eric Babson and Shmuel Onn), * Advances in Applied Math *
** 30 ** (2003) 529-544. math.CO/0207135

**[18]** Gomory Integer Programs , (with
Serkan Hosten), * Mathematical Programming Series B * **
96 ** (2003) 271 - 292.
math.OC/0106031.

**[19] **
The toric Hilbert scheme of a rank two lattice
is smooth and irreducible , (with Diane Maclagan),
* Journal of Combinatorial Theory, Series A, * ** 104 **,
Issue 1, (2003) 29--48. math.AG/0208031.

**[20]**
Symmetric iterated Betti numbers , (with Eric Babson and Isabella
Novik),* Journal of Combinatorial Theory,
Series A, * ** 105 ** (2004) 233 - 254.
math.CO/0206063.

**[21] ** Creating weekly timetables for
maximizing employee preferences , (with Caleb Z. White, Youngbae
Lee, Yoonsoo Kim and Patrick Perkins), * The UMAP Journal, *
** Vol. 25, No. 1. ** (2004) 5 - 24. More computational
tests can be seen in the older version
of the paper.

**[22] **
Toric initial ideals of $\Delta$-normal configurations:
Cohen-Macaulayness and degree bounds , (with Edwin O'Shea),
* Journal of Algebraic Combinatorics, * ** 21 **
(2005) 247 - 268. math.AC/0308109.

**[23]** The structure of group
relaxations, *
Discrete Optimization (eds: K. Aardal, G. Nemhauser, R. Weismantel),
Handbooks in Operations Research and Management Science, *
** Vol 12 **, Elsevier, 2005.

**[24]**
Reverse lexicographic and lexicographic shifting , (with Eric
Babson and Isabella Novik), * Journal of Algebraic Combinatorics,
* ** 23 ** (2006), no. 2, 107 -
123. math.CO/0507565.

**[25]** Moduli spaces of graded representations of finite
dimensional algebras, (with Eric Babson and Birge Huisgen-Zimmermann),
* Contemporary Math., * ** 419 ** (2006), 7--27.

**[26]**
Computing tropical varieties , (with Tristram Bogart, Anders Jensen,
David Speyer and Bernd Sturmfels), * Journal of Symbolic
Computation, * ** 42 ** (2007), no. 1-2, 54--73.

**[27]**
The generic Groebner walk , (with Komei Fukuda, Anders Jensen and
Niels Lauritzen), * Journal of Symbolic
Computation, * ** 43 ** (2007), no. 3, 298--312.

**[28]**
The circuit ideal of a vector configuration , (with Tristram
Bogart and Anders Jensen), * Journal of Algebra, * ** 309
** (2007), no. 2, 518--542.

**[29]** Nice initial
complexes of some classical ideals , (with Aldo Conca and Serkan
Hosten), * Contemporary Math., * ** 423 **
(2007), 11--42.

**[30]**
Computing Groebner fans , (with Komei Fukuda and Anders
N. Jensen), * Mathematics of Computation, *
** 76 ** (2007), 2189--2212.

**[31]**
Moduli of McKay quiver representations I: the coherent component ,
(with Alastair Craw and Diane Maclagan), * Proceedings
of the London Math. Society (3), * ** 95 ** (2007),
no. 1, 179--198.

**[32]**
Moduli of McKay quiver representations II: Groebner basis techniques
, (with Alastair Craw and Diane Maclagan), *
Journal of Algebra, * ** 316 ** (2007), no. 2,
514--535.

** [33] **
Theta bodies for polynomial ideals , (with João Gouveia and Pablo Parrilo),
* SIAM Journal of Optimization *
** Volume 20, Issue 4, ** (2010), 2097--2118.

Online Version at SIAM J. Opt.

** [34] **
Small Chvatal rank , (with Tristram Bogart and Annie Raymond),
* Mathematical Programming, Series B * ** Volume 124, Numbers 1-2, ** (2010)
45--68.

Online version at Springer Link

** [35] ** Convex Hulls of Algebraic Sets , (with João Gouveia), chapter in the "Handbook of Semidefinite, Cone and Polynomial Optimization", International Series in Operations Research & Management Science, Vol. 166, Miguel Anjos and Jean-Bernard Lasserre (eds), Springer, 2012.

Springer's advertisement for this book

** [36] **
A new hierarchy of semidefinite programming relaxations for cycles in binary
matroids and cuts in graphs , (with João Gouveia, Monique Laurent
and Pablo Parrilo), * Mathematical Programming, * Series A ** 133 ** (2012), no. 1-2, 203--225.

** [37] **
A Hilbert scheme in computer vision, (with Chris Aholt and Bernd Sturmfels),
* Canadian Journal of Mathematics* ** 65 ** (2013), no. 5, 961-988

** [38] **
A QCQP Approach to Triangulation, (with Chris Aholt and Sameer Agarwal),
* Computer Vision - ECCV 2012 *,
Lecture Notes in Computer Science ** 7572 ** (2012), 654-667.

** [39] **
Lifts of convex sets and cone factorizations, (with João Gouveia and Pablo Parrilo), * Mathematics of Operations Research * ** 38 ** (2013), 248-264.

** [40] **
Polytopes of minimum positive semidefinite rank,
(with João Gouveia and Richard Robinson),
*Discrete & Computational Geometry * Volume 50, Issue 3 (2013), 679-699.

** [41] **
Which nonnegative matrices are slack matrices?, (with João Gouveia,
Roland Grappe, Volker Kaibel, Kanstanstin Pashkovich and Richard Robinson),
* Linear Algebra and its Applications * 439 (2013) 2921-2933.

** [42] **
The Euclidean distance degree of an algebraic variety, (with
Jan Draisma, Emil Horobet, Giorgio Ottaviani and Bernd Sturmfels),
* Foundations of Computational Mathematics* 16:1 (2016) 99-149.

** [43] **
Approximate cone factorizations and lifts of polytopes, (with
João Gouveia and Pablo A. Parrilo),
Isaac Newton Institute preprint NI13069-POP,
* Mathematical Programming, * volume 151, Issue 2 (2015),
613-637.

** [44] **
Worst-case results for positive semidefinite rank, (with
João Gouveia and Richard Robinson),
* Mathematical Programming Series B *, Volume 153, Issue 1 (2015) 201-212.

** [45] **
Positive semidefinite rank, (with
Hamza Fawzi, João Gouveia, Pablo A. Parrilo and Richard Z. Robinson),
* Math Programming Series B *, Volume 153, Issue 1 (2015) 133-177.

** [46] **
Counting real critical points of the distance to spectral matrix sets, (with
Dimitriy Drusvyatskiy and Hon-Leung Lee), * SIAM J. Matrix Analysis and Applications *, Volume 36, Issue 3 (2015), pp. 1360-1380.

** [47] **
Four dimensional polytopes of minimum positive semidefinite rank, (with
João Gouveia, Kanstanstin Pashkovich and Richard Z. Robinson),
* J. Combin. Theory Ser. A * 145 (2017), 184--226.
Website with all the computations in this paper

** [48] **
The Euclidean distance degree of orthogonally invariant matrix varieties, (with
Dimitriy Drusvyatskiy, Hon-Leung Lee and Giorgio Ottaviani),
to appear in the
* Israel Journal of Mathematics. *

** [49] **
On the existence of epipolar matrices, (with
Sameer Agarwal, Hon-Leung Lee and Bernd Sturmfels),
* Int. J. Comput. Vis. * 121 (2017), no. 3, 403--415

** [50] **
Symmetric sums of squares over k-subset hypercubes
(with Annie Raymond, James Saunderson and Mohit Singh),
to appear in * Mathematical Programming * Series A.

### Only on the Math arXiv

Certifying the existence of epipolar matrices, (with
Sameer Agarwal, Hon-Leung Lee and Bernd Sturmfels),
July 2014.