Submitted:
16 September 2025
Posted:
18 September 2025
Read the latest preprint version here
Abstract
Keywords:
MSC: Primary: 32SO5; Secondary: 13C15
1. Introduction
- 1968-2
- What topological characteristics of a real (complex) polynomial are computable from the Newton polyhedron (and the signs of the coefficients)?
- 1975-1
- Every interesting discrete invariant of a generic singularity with a Newton polyhedron is an interesting function of the polyhedron. Study: the signature, the number of moduli, the singularity index, the integral monodromy, the variation, the Bernstein polynomial, and (for generic section).
- 1975-21
- Express the main numerical invariants of a typical singularity with a given Newton polyhedron (e.g., the signature, the genus of the 1-dimensional Milnor fiber) in terms of the polyhedron.
2. Preliminaries

- - support off
- - convex hull of - Newton polyhedron of f
- - family of compact faces of - Newton boundary of f
- f - Kushnirenko nondegenerate on if the system of equationshas no solution in .
- f - Kushnirenko nondegenerate, if f nondegenerate on each face
- satisfies k-condition and it does not satisfy - condition.
3. Main Results
- (i)
- (ii)
-
satisfies d-condition and it does not satisfy - condition,for each
4. The Proof of the Main Results
- Then Put Then are empty sets. Hence does not satisfy (1) - Kushnirenko condition.
-
Since f is nondegenerate, then is also nondegenerate andTherefore without loss of a generality we can assume thatNow, let’s expand and with respect toHence Therefore function are identically equal to 0 and f has a form:or Put Then are empty sets. Hence does not satisfy (1) - Kouchnirenko condition.
Funding
Conflicts of Interest
References
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