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. The volume polynomial of lattice polygons (with J. Soprunova) preprint, see GitHub for code
  2. The poset of binary CSS-T quantum codes and cyclic codes (with E. Camps, H. López, G. Matthews, D. Ruano, and R. San-José) preprint
  3. Relative hulls and quantum codes (with S. Anderson, E. Camps, H. López, G. Matthews, and D. Ruano) preprint
  4. 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
  5. 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
  6. 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
  7. The dual of an evaluation code (with H. López and R. Villarreal) Designs, Codes and Cryptography, 89, (2021) 1367-1403
  8. 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
  9. Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum (with G. Averkov and C. Borger) Mathematika 66 (2020) 1003-1027
  10. 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
  11. Wulff shapes and a characterization of simplices via a Bezout type inequality (with C. Saroglou and A. Zvavitch) Advances in Mathematics 357 (2019) 106789
  12. Criteria for strict monotonicity of the mixed volume of convex polytopes (with F. Bihan) Advances in Geometry, vol 19, issue 4 (2019) 527-540
  13. Generalized multiplicities of edge ideals (with A. Alilooee and J. Validashti) Journal of Algebraic Combinatorics, vol 47, issue 3 (2018) 441-472 
  14. 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
  15. Bezout Inequality for Mixed volumes (with A. Zvavitch) International Math Research Notices , 23 (2016) 7230-7252
  16. Eventual quasi-linearity of the Minkowski length (with J. Soprunova) European Journal of Combinatorics 58 (2016) 107-117
  17. Multigraded Hilbert function and toric complete intersection codes (with M. Şahin) Journal of Algebra 459 (2016) 446-467
  18. Lattice polytopes in coding theory Journal of Algebra Combinatorics Discrete Structures and Applications, vol 2, no 2 (2015) 85-94
  19. Tropical determinant on transportation polytope (with S. Gajula and J. Soprunova) Linear Algebra and its Applications, vol 475, no 15 (2015) 28-44
  20. On dual toric complete intersection codes (with P. Celebi Demirarslan) Finite Fields and Their Applications, vol 33 (2015) 118-136
  21. Toric complete intersection codes Journal of Symbolic Computation, 50 (2013) 374-385
  22. Bringing toric codes to the next dimension (with J. Soprunova) SIAM J. Discrete Math. 24, no 2, (2010) 655-665
  23. Toric surface codes and Minkowski length of polygons (with J. Soprunova) SIAM J. Discrete Math. 23, no 1 (2009) 384-400
  24. 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)
  25. Global residues for sparse polynomial systems Journal of Pure and Applied Algebra 209, no 2, (2007) 383-392
  26. Combinatorial construction of toric residues (with A. Khetan) Ann. Inst. Fourier (Grenoble) 55, no 2 (2005) 511-548
  27. Toric residue and combinatorial degree Trans. Amer. Math. Soc. 357, no 5 (2005) 1963-1975
  28. Residues and tame symbols on toroidal varieties Compositio Math. 140, no 6 (2004) 1593-1613
  29. On combinatorial coefficients and the Gelfond-Khovanskii residue formula Topics in Algebraic Geometry and Geometric Modeling, Contemp. Math., 334, AMS (2003) 343-349
  30. Parshin’s symbols and residues, and Newton polyhedra Ph.D. Thesis, University of Toronto (2002)
  31. Symbols and residues on surfaces preprint (2001)
  32. 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”
  33. Patching theorem in quadratic Serre’s conjecture Fundam. Prikl. Mat. 3, no 1 (1997) 255-261 (in Russian)
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