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Condition (I) conjecture for the elementary divisors of quasihomogeneous singularities

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Let f∈C[x1,…,xn]f\in\mathbb{C}[x_1,\ldots,x_n] be a quasihomogeneous singularity. Define the finite sets M1⊃⋯⊃Mνmax⁡M_1\supset\cdots\supset M_{\nu_{\max}} from the multiplicities of the cyclotomic factors in the characteristic polynomial of its monodromy. For a finite set MM of positive integers, let G(M){\mathcal G}(M) be the graph whose vertices are the elements of MM, with an edge from m1m_1 to m2m_2 when m1/m2m_1/m_2 is a power of a prime. Condition (I) means that G(M){\mathcal G}(M) is connected, satisfies (S2)(S_2), and satisfies (Tp)(T_p) for every prime p≥3p\geq3. Condition (I) conjecture. For any quasihomogeneous singularity, each of the sets M1,…,Mνmax⁡M_1,\ldots,M_{\nu_{\max}} satisfies condition (I). This conjecture is presented as an amendment to Orlik's conjecture and is posed as an open problem in the source.

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  1. Condition (I) conjecture for the elementary divisors of quasihomogeneous singularities

    Let f∈C[x1,…,xn]f\in\mathbb{C}[x_1,\ldots,x_n] be a quasihomogeneous singularity, and let M1,…,Mνmax⁡M_1,\ldots,M_{\nu_{\max}} be the finite sets obtained from the cyclotomic multiplicities of its monodromy characteristic polynomial. For a finite set MM of positive integers, condition (I) is the graph-theoretic condition that G(M){\mathcal G}(M) is connected, satisfies (S2)(S_2), and satisfies (Tp)(T_p) for every prime p≥3p\geq3. Condition (I) conjecture. For any quasihomogeneous singularity, each of the sets M1,…,Mνmax⁡M_1,\ldots,M_{\nu_{\max}} satisfies condition (I). The claim is the same amendment to Orlik's conjecture stated earlier in the paper; the source subsequently refers to it as an open problem, despite the parser's disproved status marker.

    source: Claus Hertling and Philip Zilke, “Seven combinatorial problems around quasihomogeneous singularities”, arXiv:1801.08272 (2018).

References

Primary source

Claus Hertling and Philip Zilke, “Seven combinatorial problems around quasihomogeneous singularities”, arXiv:1801.08272 (2018).

combinatoricsquasihomogeneous-singularitiessingularity-theory
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