My research is in combinatorial and computational algebraic geometry. I am particularly interested in toric varieties, their applications to sparse polynomial systems, combinatorics and geometry of polyhedra, and coding theory. Recently, I have been also working on geometric inequalities involving mixed volumes.

Publications

  1. An algebraic-combinatorial proof of a Bezout-type inequality for mixed volumes of three-dimensional zonoids (with G. Averkov), preprint
  2. Binary Triorthogonal and CSS-T Codes for Quantum Error Correction (with E. Camps, H. López, G. Matthews, D. Ruano, and R. San-José), preprint
  3. On the affine permutation group of certain decreasing Cartesian codes (with E. Camps, H. López, and E. Sarmiento), preprint
  4. Mixed volumes of zonoids and the absolute value of the Grassmannian (with G. Averkov, K. von Dichter, and S. Richard), preprint
  5. The volume polynomial of lattice polygons (with J. Soprunova) Proceedings of AMS, vol 152, no 12, (2024), 5313-5325, see GitHub for code
  6. An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance (with E. Camps, H. López, G. Matthews, D. Ruano, and R. San-José) Quantum Information Processing, vol 23, article number 230, (2024)
  7. Relative hulls and quantum codes (with S. Anderson, E. Camps, H. López, G. Matthews, and D. Ruano) IEEE Transactions on Information Theory, (2024), vol 70, issue 5, 3190-3201
  8. Plücker-type inequalities for mixed areas and intersection numbers of curve arrangements (with G. Averkov) International Math Research Notices, vol 2023, issue 18, (2023) 16015–16050
  9. 𝔽q-zeros of sparse trivariate polynomials and toric 3-fold codes (with K. Meyer and J. Soprunova) SIAM Applied Algebra and Geometry, vol 6, issue 3 (2022), see also GitHub for code
  10. Coding theory package for Macaulay2 (with T. Ball, E. Camps, H. Chimal-Dzul, D. Jaramillo-Velez, H. López, N. Nichols, M. Perkins, G. Vera-Martínez, and G. Whieldon) Journal of Software for Algebra and Geometry, vol 11 (2021) 117-122
  11. The dual of an evaluation code (with H. López and R. Villarreal) Designs, Codes and Cryptography, 89, (2021) 1367-1403
  12. Classification of triples of lattice polytopes with a given mixed volume (with G. Averkov and C. Borger) Discrete & Computational Geometry, 66, (2021) 165-202 , see also GitHub for data and code
  13. Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum (with G. Averkov and C. Borger) Mathematika 66 (2020) 1003-1027
  14. Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes (with H. López and G. Matthews) Designs, Codes and Cryptography, 88, no 8, (2020) 1673-1685
  15. Wulff shapes and a characterization of simplices via a Bezout type inequality (with C. Saroglou and A. Zvavitch) Advances in Mathematics 357 (2019) 106789
  16. Criteria for strict monotonicity of the mixed volume of convex polytopes (with F. Bihan) Advances in Geometry, vol 19, issue 4 (2019) 527-540
  17. Generalized multiplicities of edge ideals (with A. Alilooee and J. Validashti) Journal of Algebraic Combinatorics, vol 47, issue 3 (2018) 441-472 
  18. Characterization of simplices via the Bezout Inequality for Mixed volumes (with C. Saroglou and A. Zvavitch) Proceedings of AMS, vol 144, no 12, (2016) 5333-5340
  19. Bezout Inequality for Mixed volumes (with A. Zvavitch) International Math Research Notices , 23 (2016) 7230-7252
  20. Eventual quasi-linearity of the Minkowski length (with J. Soprunova) European Journal of Combinatorics 58 (2016) 107-117
  21. Multigraded Hilbert function and toric complete intersection codes (with M. Şahin) Journal of Algebra 459 (2016) 446-467
  22. Lattice polytopes in coding theory Journal of Algebra Combinatorics Discrete Structures and Applications, vol 2, no 2 (2015) 85-94
  23. Tropical determinant on transportation polytope (with S. Gajula and J. Soprunova) Linear Algebra and its Applications, vol 475, no 15 (2015) 28-44
  24. On dual toric complete intersection codes (with P. Celebi Demirarslan) Finite Fields and Their Applications, vol 33 (2015) 118-136
  25. Toric complete intersection codes Journal of Symbolic Computation, 50 (2013) 374-385
  26. Bringing toric codes to the next dimension (with J. Soprunova) SIAM J. Discrete Math. 24, no 2, (2010) 655-665
  27. Toric surface codes and Minkowski length of polygons (with J. Soprunova) SIAM J. Discrete Math. 23, no 1 (2009) 384-400
  28. Let me tell you my favorite lattice point problem… (with M. Beck, B. Nill, B. Reznick, C. Savage, Z. Xu) Integer points in polyhedra, Contemp. Math. 452, AMS (2008)
  29. Global residues for sparse polynomial systems Journal of Pure and Applied Algebra 209, no 2, (2007) 383-392
  30. Combinatorial construction of toric residues (with A. Khetan) Ann. Inst. Fourier (Grenoble) 55, no 2 (2005) 511-548
  31. Toric residue and combinatorial degree Trans. Amer. Math. Soc. 357, no 5 (2005) 1963-1975
  32. Residues and tame symbols on toroidal varieties Compositio Math. 140, no 6 (2004) 1593-1613
  33. On combinatorial coefficients and the Gelfond-Khovanskii residue formula Topics in Algebraic Geometry and Geometric Modeling, Contemp. Math., 334, AMS (2003) 343-349
  34. Parshin’s symbols and residues, and Newton polyhedra Ph.D. Thesis, University of Toronto (2002)
  35. Symbols and residues on surfaces preprint (2001)
  36. A short proof of the Prime Number Theorem for arithmetic progressions (1998) This paper was written in response to Don Zagier’s paper “Newman’s Short Proof of the Prime Number Theorem”
  37. Patching theorem in quadratic Serre’s conjecture Fundam. Prikl. Mat. 3, no 1 (1997) 255-261 (in Russian)
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