Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

Estimating the number of solutions of polynomial systems of equations in terms of Newton polytopes, in 1974 the author proved that the codimension of the ideal (g1, g2, . . . , gd) generated in the group algebra K[Zd] over the field K of characteristic 0 by Laurent polynomials of general position having the same Newton polytope Γ is equal to d! × V olume(Γ). Assuming that the Newton polyhedron is simplicial and super-convenient (i.e. containing some neighborhood of the origin), the author re-proves and strengthens the 1974 result by explicitly indicating the set Bsh of monomials whose equivalence classes form a basis for the quotient algebra K[Zd]/(g1, g2, . . . , gd). It is proved that the cardinality of this set is equal to d! × V olume(Γ). By a well-known theorem of commutative algebra, it follows that in the case of an algebraically closed field K of characteristic 0, the number of solutions of the system of equations g1 = g2 = . . . = gd = 0, taking into account multiplicities, will be equal to d! × V olume(Γ). The set Bsh has an analogue of the Dehn-Sommerville property and arises naturally in the process of calculating the Poincar´e series of the linear space of Laurent polynomials equipped with the Arnold- Newton grading. The inductive construction of the set Bsh relies on the construction of the shelling sh whose existence for any convex polyhedron was proved in 1971 by Bruggerser and Money. Using the structure of Bsh, we prove that the associated graded K-algebra grΓ(K[Zd]) constructed from the Arnold-Newton piecewise linear filtration of the K-algebra K[Zd] has the Cohen-Macaulay property. Our proof of the Cohen-Macaulay property is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof of the Cohen-Macaulay property of the Stanley-Reisner rings of simplicial complexes admitting shelling. Using the Cohen-Macaulay property of grΓ(K[Zd]), we prove that for generic Laurent polynomials (g1, g2, . . . , gd) that have the same Newton polytope Γ, the set Bsh is a monomial basis of the quotient algebra K[Zd]/(g1, g2, . . . , gd). The results of the paper can easily be extended to ordinary polynomials and formal series, which will be the subject of a separate publication.

1. В. И. Арнольд, Нормальные формы функций в окрестности вырожденных критических точек, УМН, 29:2(176) (1974), с. 11–49; Russian Math. Surveys, 29:2 (1974), pp.10–50 https://www.mathnet.ru/links/803374d14c3fdbe7d67529c757f7fba3/rm4352.pdf

2. А.Г. Кушниренко, Многогранник Ньютона и числа Милнора, Функц. анализ, том 9, вып. 1 (1975), с. 74-75 https://www.mathnet.ru/links/329399444a7b795f6fd30180e0ff2644/faa2223.pdf, DOI: https://doi.org/10.1007/BF01078188

3. А.Г. Кушниренко, Многогранники Ньютона и теорема Безу, Функц. анализ, том 10, вып. 3 (1976), с. 82-83 https://www.mathnet.ru/links/5744b3b4585c5707364a9649a3845d9a/faa2179.pdf, DOI:https://doi.org/10.1007/BF01075534

4. A.G. Kouchnirenko, Poly`edres de Newton et nombres de Milnor, Inventiones mathematicae 32 (1976) pp. 1-32 https://link.springer.com/content/pdf/10.1007/BF01389769.pdf

5. M.Hochster, Rings Invariants of Tori, Cohen-Macaulay Rings Generated by Monomials, and Polytopes, Annals of Mathematics Second Series, Vol. 96, No. 2 (Sep., 1972), pp. 318-337, DOI: https://doi.org/10.2307/1970791

6. M.Hochster, Cohen-Macaulay Varieties, Geometric Complexes, and Combinatorics, http://www.math.lsa.umich.edu/~hochster/comb2.pdf

7. Merle, M.. Les anneaux coniques sont de Cohen-Macaulay, d’apr`es A.G. Kouchnirenko. S´eminaire sur les singularit´es des surfaces (1976-1977): 1-7. http://eudml.org/doc/114151

8. Anatoly Kushnirenko, Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra, Mathematics 2022, 10(23), 4445; https://doi.org/10.3390/math10234445

9. Э. Б. Винберг, М. Джибладзе, А. Г. Элашвили, Алгебры модулей некоторых неполуквазиоднородных особенностей, Функц. анализ и его прил., 51:2 (2017), 10–24; Funct. Anal. Appl., 51:2 (2017), 86–97 https://doi.org/10.4213/faa3450 https://doi.org/10.1007/s10688-017-0171-6

10. B. Kind and P. Kleinschmidt, Sch¨albare Cohen–Macaulay Komplexe und ihre Parametrisierung, Math. Z. 167(1979), pp. 173-179, https://eudml.org/doc/172845

11. Richard P. Stanley, A glimpse of combinatorial commutative algebra, www-math.mit.edu/~rstan/algcomb/chapter13.pdf

12. Richard P. Stanley, Combinatorics and commutative algebra. Birkh¨auser, 1983

13. H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres, Math. Scand., 29 (1971), pp. 197-205, https://www.jstor.org/stable/24491028

14. Jean Gallier. Notes on Convex Sets, Polytopes, PolyhedraCombinatorial Topology, Voronoi Diagrams and Delaunay Triangulations. [Research Report], RR-6379, INRIA. 2007, pp.179, https://hal.inria.fr/inria-00193831v3

15. Arnold, V. I., Arnold’s problems, Springer-Verlag, Berlin; PHASIS, Moscow (2004), http://www.vixri.ru/d2/Arnold%20V%20_Arnolds%20Problems%20,%20653str.pdf

16. Brzostowski, S., Krasi´nski, T., Walewska, J., Arnold’s problem on monotonicity of the Newton number for surface singularities. J. Math. Soc. Japan 71:4 (2019), 1257–1268, arXiv:1705.00323

17. Fedor Selyanin, A non-negative analogue of the Kouchnirenko formula (2020), https://arxiv.org/pdf/2006.11795v1.pdf

18. Fedor Selyanin, Arnold’s monotonicity problem, https://arxiv.org/pdf/2006.11795v3.pdf

19. Stapledon, A. Formulas for monodromy. Res Math Sci 4, 8 (2017). https://doi.org/10.1186/s40687-017-0097-x

20. Maximiliano Leyton- ´Alvarez, Hussein Mourtada, Mark Spivak, Newton non-degenerate μ- constant deformations admit simultaneous embedded resolutions, Compositio Mathematica, Volume 158 , Issue 6 , June 2022 , pp. 1268 – 1297 DOI: https://doi.org/10.1112/S0010437X22007576

21. Maximiliano Leyton- ´Alvarez, Hussein Mourtada, Mark Spivak, Newton non-degenerate μ-constant deformations admit simultaneous embedded resolutions, v5, arXiv:2001.10316v5, 2024, https://arxiv.org/abs/2001.10316v5

22. Alexander Barvinok, Lattice Points, Polyhedra, and Compexity. Lecture 3, Theorem 1, and Theorem 2, in Geometric Combinatorics / Ezra Miller, Victor Reiner, Bernd Sturmfels, editors. – IAS/ParkCity mathematics series, ISSN 1079-5634; v. 13, American Mathematical Soc., 2007. http://www.math.lsa.umich.edu/ barvinok/lectures.pdf

23. G¨unter M. Ziegler, Lectures on Polytopes (in Graduate Texts in Mathematics, Vol 152), Springer- Verlag New York, 1995.

24. Douai, A., Sabbah, C.: Gauss–Manin systems, Brieskorn lattices and Frobenius structures I. Ann. Inst. Fourier 53(4), 1055–1116 (2003), https://doi.org/10.48550/arXiv.math/0211352

25. Douai, A.: A note on the Newton spectrum of a polynomial. arXiv:1810.03901, https://doi.org/10.48550/arXiv.1810.03901

26. Douai, A. Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials. J Algebr Comb 54, 719–732 (2021). https://doi.org/10.1007/s10801-020-00984-x

27. A. Stapledon, Weighted Ehrhart theory and orbifold cohomology, Advances in Mathematics, Volume 219, Issue 1,10 September 2008, Pages 63-88, DOI: https://doi.org/10.1016/j.aim.2008.04.010

28. Stanley, R.P., Combinatorics and Commutative Algebra, 2nd ed.; Progress in Mathematics, 41; Birkh¨auser Boston, Inc.: Boston, MA, USA, 1996; ISBN 0-8176-3836-9.

29. Атья М., Макдональд И., Введение в коммутативную алгебру. "Мир", М.- 1972. https://ikfia.ysn.ru/wp-content/uploads/2018/01/AtjaMakdonald1972ru.pdf M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass. (1969). xx IX+128 pp. DOI: https://doi.org/10.1017/S0008439500031039

30. M. Hochster, Cohen-Macoley rings, http://www.math.lsa.umich.edu/~hochster/615W14/CM.pdf

31. Н. Бурбаки, Коммутативная алгебра, М.: Мир, 1971, 707 стр. https://ikfia.ysn.ru/wp-content/uploads/2018/01/Burbaki1971ru.pdf N. Bourbaki, Commutative Algebra, Chapters 1-7, Springer-Verlag, New York, 1985.

32. A. Ogus, Commutative Algebra, https://math.berkeley.edu/~ogus/Math_250B-2016/Notes/koszul.pdf