### Preprints

Projectively unique polytopes and
toric slack ideals,
August 2018.

Spectrahedral lifts of convex sets,
March 2018.

On the local stability of semidefinite relaxations,
(with Diego Cifuentes, Sameer Agarwal and Pablo A. Parrilo)
October 2017.

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

### 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),
* Israel Journal of Mathematics,* 221 (2017), no. 1, 291--316.

** [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),
* Mathematical Programming * 167 (2018), no.2, Ser. A, 315--354.

** [51] **
Symmetry in Turán sums of squares polynomials from flag algebras,
(with Annie Raymond and Mohit Singh), * Algebraic Combinatorics * 1 (2018), no.2, 249--274.

### Only on the Math arXiv

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