Abstract
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions.
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References
Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)
Bürgisser, P., Scheiblechner, P.: On the complexity of counting components of algebraic varieties. J. Symb. Comput. 44(9), 1114–1136 (2009)
Cattani, E., Dickenstein, A.: Counting solutions to binomial complete intersections. J. Complexity 23(1), 82–107 (2007)
Cox, D.A., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185. New York: Springer (1998)
Cueto, M.A., Dickenstein, A.: Some results on inhomogeneous discriminants. In: Ferrer Santos, W. et al. (eds.) Proceedings of the XVIth Latin American Algebra Colloquium. Biblioteca de la Revista Matemática Iberoamericana, pp. 41–62. Revista Matemática Iberoamericana, Madrid (2007)
Dayton, B.H., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (ISSAC’05), pp. 116–123. ACM, New York (2005)
Emiris, I.Z., Verschelde, J.: How to count efficiently all affine roots of a polynomial system. Discrete Appl. Math. 93(1), 21–32 (1999)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994)
Herrero, M.I., Jeronimo, G., Sabia, J.: Computing isolated roots of sparse polynomial systems in affine space. Theor. Comput. Sci. 411(44–46), 3894–3904 (2010)
Herrero, M.I., Jeronimo, G., Sabia, J.: Affine solution sets of sparse polynomial systems. J. Symb. Comput. 51, 34–54 (2013)
Huber, B., Sturmfels, B.: Bernstein’s theorem in affine space. Discrete Comput. Geom. 17(2), 137–141 (1997)
Kaveh, K., Khovanskii, A.G.: Convex bodies and multiplicities of ideals. Proc. Steklov Inst. Math. 286(1), 268–284 (2014)
Khovanskii, A.G.: Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11(4), 289–296 (1978)
Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1–31 (1976)
Kushnirenko, A.G.: Newton polytopes and the Bézout theorem. Funct. Anal. Appl. 10(3), 233–235 (1976)
Li, T.Y., Wang, X.: The BKK root count in \(\mathbb{C}^n\). Math. Comput. 65(216), 1477–1484 (1996)
Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1916)
Mondal, P.: Intersection multiplicity, Milnor number and Bernstein’s theorem (2016). arXiv:1607.04860
Oka, M.: Non-Degenerate Complete Intersection Singularity. Actualités Mathématiques. Hermann, Paris (1997)
Philippon, P., Sombra, M.: Hauteur normalisée des variétés toriques projectives. J. Inst. Math. Jussieu 7(2), 327–373 (2008)
Philippon, P., Sombra, M.: A refinement of the Bernštein–Kušnirenko estimate. Adv. Math. 218(5), 1370–1418 (2008)
Rojas, J.M.: A convex geometric approach to counting the roots of a polynomial system. Theor. Comput. Sci. 133(1), 105–140 (1994)
Rojas, J.M., Wang, X.: Counting affine roots of polynomial systems via pointed Newton polytopes. J. Complexity 12(2), 116–133 (1996)
Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004)
Teissier, B.: Monômes, volumes et multiplicités. In: Tráng, L.D. (ed.) Introduction à laThéorie des Singularités, II. Travaux en Cours, vol. 37, pp. 127–141. Hermann, Paris (1988)
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The authors wish to thank the referees for their careful reading and helpful suggestions.
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Editor in Charge: Kenneth Clarkson
Partially supported by the following Argentinian grants: PIP 11220130100527CO CONICET (2014-2016) and UBACYT 2017, 20020160100039BA.
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Herrero, M.I., Jeronimo, G. & Sabia, J. On the Multiplicity of Isolated Roots of Sparse Polynomial Systems. Discrete Comput Geom 62, 788–812 (2019). https://doi.org/10.1007/s00454-018-0025-x
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DOI: https://doi.org/10.1007/s00454-018-0025-x