Abstract
We present a number of observations concerning the so-called invertible polynomials introduced and studied in a series of papers on mathematical physics and singularity theory. Specifically, we consider real versions of invertible polynomials and investigate invariants of the associated isolated hypersurface singularities. By the very definition, such a polynomial is weighted homogeneous and its gradient vector field \(\operatorname{grad}f\) has an isolated zero at the origin; hence its index \(\operatorname{ind}_0\operatorname{grad}f\) is well defined. This index, referred to as the gradient index of the polynomial, is our main concern. In particular, we give an effective estimate for the absolute value of the gradient index \(\operatorname{ind}_0\operatorname{grad}f\) in terms of the weighted homogeneous type of \(f\) and investigate its sharpness. For real invertible polynomials in two and three variables, we give the whole set of possible values of the gradient index. As an application, in the case of three variables we give a complete list of possible topological types of Milnor fibres of real invertible polynomials, which generalizes recent results of L. Andersen on the topology of isolated real hypersurface singularities. In conclusion we present a few open problems and conjectures suggested by our results.
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References
L. Andersen, “On isolated real singularities. I,” arXiv: 2110.04407 [math.AG].
L. Andersen, “On isolated real singularities. II,” arXiv: 2110.04417 [math.AG].
V. I. Arnol’d, “Index of a singular point of a vector field, the Petrovskii–Oleinik inequality, and mixed Hodge structures,” Funct. Anal. Appl. 12 (1), 1–12 (1978) [transl. from Funkts. Anal. Prilozh. 12 (1), 1–14 (1978)].
V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1: Classification of Critical Points, Caustics and Wave Fronts; Vol. 2: Monodromy and Asymptotics of Integrals (Birkhäuser, Boston, 2012) [transl. from Russian (MTsNMO, Moscow, 2012)].
P. Berglund and T. Hübsch, “A generalized construction of mirror manifolds,” Nucl. Phys. B 393 (1–2), 377–391 (1993).
W. Ebeling and S. M. Gusein-Zade, “Orbifold Euler characteristics for dual invertible polynomials,” Moscow Math. J. 12 (1), 49–55 (2012).
W. Ebeling and A. Takahashi, “Strange duality of weighted homogeneous polynomials,” Compos. Math. 147 (5), 1413–1433 (2011).
D. Eisenbud and H. I. Levine, “An algebraic formula for the degree of a \(C^{\infty }\) map germ,” Ann. Math., Ser. 2, 106, 19–44 (1977).
G. K. Giorgadze and G. N. Khimshiashvili, “Nondegeneracy of certain constrained extrema,” Dokl. Math. 92 (3), 691–694 (2015) [transl. from Dokl. Akad. Nauk 465 (3), 269–273 (2015)].
S. M. Gusein-Zade, “On the existence of deformations without critical points (the Teissier problem for functions of two variables),” Funct. Anal. Appl. 31 (1), 58–60 (1997) [transl. from Funkts. Anal. Prilozh. 31 (1), 74–77 (1997)].
N. Hussain, S. S.-T. Yau, and H. Zuo, “On the derivation Lie algebras of fewnomial singularities,” Bull. Aust. Math. Soc. 98 (1), 77–88 (2018).
G. N. Khimshiashvili, “On the local degree of a smooth mapping,” Soobshch. Akad. Nauk Gruz. SSR 85 (2), 309–312 (1977).
G. Khimshiashvili, “Signature formulae for topological invariants,” Proc. A. Razmadze Math. Inst. 125, 1–121 (2001).
G. Khimshiashvili, “New applications of algebraic formulae for topological invariants,” Georgian Math. J. 11 (4), 759–770 (2004).
A. G. Khovanskii, “Index of a polynomial vector field,” Funct. Anal. Appl. 13 (1), 38–45 (1979) [transl. from Funkts. Anal. Prilozh. 13 (1), 49–58 (1979)].
A. G. Khovanskii, Fewnomials (Am. Math. Soc., Providence, RI, 1991), Transl. Math. Monogr. 88.
M. Kreuzer, “The mirror map for invertible LG models,” Phys. Lett. B 328 (3–4), 312–318 (1994).
M. Kreuzer and H. Skarke, “On the classification of quasihomogeneous functions,” Commun. Math. Phys. 150 (1), 137–147 (1992).
D. W. Masser, “The discriminants of special equations,” Math. Gaz. 50 (372), 158–160 (1966).
J. N. Mather and S. S.-T. Yau, “Classification of isolated hypersurface singularities by their moduli algebras,” Invent. Math. 69, 243–251 (1982).
J. W. Milnor, Singular Points of Complex Hypersurfaces (Princeton Univ. Press, Princeton, NJ, 1968), Ann. Math. Stud. 61.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 321, pp. 94–107 https://doi.org/10.4213/tm4317.
Dedicated to Academician Revaz Valerianovich Gamkrelidze with sincere respect and gratitude
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Giorgadze, G., Khimshiashvili, G. On the Index of the Gradient of a Real Invertible Polynomial. Proc. Steklov Inst. Math. 321, 84–96 (2023). https://doi.org/10.1134/S0081543823020062
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DOI: https://doi.org/10.1134/S0081543823020062