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Polar degrees and closest points in codimension two

    Suppose that XAn1 is a toric variety of codimension two defined by an (n2)×n integer matrix A, and let B be a Gale dual of A. In this paper, we compute the Euclidean distance degree and polar degrees of XA (along with other associated invariants) combinatorially working from the matrix B. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix A in the codimension two case.

    Communicated by J. Hauenstein

    AMSC: 14M25, 14Q20, 52B20, 52B35


    • 1. M. F. Adamer and M. Helmer, Euclidean distance degree for chemical reaction networks, preprint (2017), arXiv:1707.07650. Google Scholar
    • 2. P. Aluffi, Projective duality and a Chern-Mather involution, Trans. Amer. Math. Soc. 370(3) (2018) 1803–1822. Crossref, ISIGoogle Scholar
    • 3. B. Bank, M. Giusti, J. Heintz, M. S. El Din and E. Schost, On the geometry of polar varieties, Appl. Algebra Eng. Commun. Comput. 21 (1) (2010) 33–83. Crossref, ISIGoogle Scholar
    • 4. B. Büeler, A. Enge and K. Fukuda, Exact volume computation for polytopes: a practical study, in Polytopes — Combinatorics and Computation (Springer, 2000), pp. 131–154. CrossrefGoogle Scholar
    • 5. B. Chazelle, An optimal convex hull algorithm in any fixed dimension, Discrete Comput. Geom. 10(1) (1993) 377–409. Crossref, ISIGoogle Scholar
    • 6. D. A. Cox, J. B. Little and H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics, Vol. 124 of (American Mathematical Society, Providence, RI, 2011). CrossrefGoogle Scholar
    • 7. A. Dickenstein and B. Sturmfels, Elimination theory in codimension 2, J. Symbolic Comput. 34(2) (2002) 119–135. Crossref, ISIGoogle Scholar
    • 8. J. Draisma, E. Horobeţ, G. Ottaviani, B. Sturmfels and R. R. Thomas, The Euclidean distance degree of an algebraic variety, Found. Comput. Math. 16(1) (2016) 99–149. Crossref, ISIGoogle Scholar
    • 9. M. E. Dyer and A. M. Frieze, On the complexity of computing the volume of a polyhedron, SIAM J. Comput. 17(5) (1988) 967–974. Crossref, ISIGoogle Scholar
    • 10. W. Fulton, Intersection Theory, Vol. 2 (Springer Science & Business Media, 2013). Google Scholar
    • 11. I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications (Birkhäuser Boston, Boston, MA, 1994). CrossrefGoogle Scholar
    • 12. D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry (2017). Google Scholar
    • 13. M. Helmer and B. Sturmfels, Nearest points on toric varieties, preprint (2016), arXiv:1603.06544. Google Scholar
    • 14. A. Holme, The geometric and numerical properties of duality in projective algebraic geometry, Manuscripta Mathematica 61(2) (1988) 145–162. Crossref, ISIGoogle Scholar
    • 15. M. E. Keating, A First Course in Module Theory (World Scientific, 1998). LinkGoogle Scholar
    • 16. S. L. Kleiman et al., Tangency and duality, in Proc. 1984 Vancouver Conf. Algebraic Geometry, Vol. 6, (1986), pp. 163–225. Google Scholar
    • 17. F. C. O. Los, T. M. Randis, R. V. Aroian and A. J. Ratner, Role of pore-forming toxins in bacterial infectious diseases, Microbiol. Mol. Biol. Rev. 77(2) (2013) 173–207. Crossref, ISIGoogle Scholar
    • 18. R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974) 423–432. Crossref, ISIGoogle Scholar
    • 19. Y. Matsui and K. Takeuchi, A geometric degree formula for A-Discriminants and Euler obstructions of toric varieties, Adv. Math. 226(2) (2011) 2040–2064. Crossref, ISIGoogle Scholar
    • 20. E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, Vol. 227 (Springer Science & Business Media, 2004). Google Scholar
    • 21. B. I. U. Nødland, Local Euler obstructions of toric varieties, J. Pure Appl. Algebra 222(3) (2018) 508–533. CrossrefGoogle Scholar
    • 22. L. O’Carroll, F. Planas-Vilanova and R. H. Villarreal, Degree and algebraic properties of lattice and matrix ideals, SIAM J. Discrete Math. 28(1) (2014) 394–427. Crossref, ISIGoogle Scholar
    • 23. G. Ottaviani, P.-J. Spaenlehauer and B. Sturmfels, Exact solutions in structured low-rank approximation, SIAM J. Matrix Anal. Appl. 35(4) (2014) 1521–1542. Crossref, ISIGoogle Scholar
    • 24. R. Piene, Polar classes of singular varieties, Ann. Sci. École Norm. Sup. (4) 11(2) (1978) 247–276. CrossrefGoogle Scholar
    • 25. R. Piene, Polar varieties revisited, in Computer Algebra and Polynomials, Lecture Notes in Computer Science, Vol. 8942 (Springer, Cham, 2015), pp. 139–150. CrossrefGoogle Scholar
    • 26. R. Piene, Chern–Mather classes of toric varieties, preprint (2016), arXiv:1604.02845. Google Scholar
    • 27. J. J. Rotman, Advanced Modern Algebra Graduate Studies in Mathematics, Vol. 114 (American Mathematical Society, Providence, RI, 2010). CrossrefGoogle Scholar
    • 28. E. A. Tevelev, Projective Duality and Homogeneous Spaces, Encyclopardia of Mathematical Sciences, Vol. 133 (Springer Science & Business Media, 2006). Google Scholar