Abstract
Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over \(\mathbb {C}\), this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.
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Notes
Our results still hold under the weaker assumption that the characteristic of \(\mathbb {K}\) does not divide \({d}\).
As usual, the notation \({{\mathcal {O}}}^{\sim}()\) hides logarithmic factors
The terminology pseudo-irreducible is also used in [14] to design polynomials which cannot be factored into comaximal polynomials. These two notions are not related to each other.
In the sequel, we rather use the terminology balanced and give an alternative definition of pseudo-irreducibility based on a Newton–Puiseux type algorithm. Both notions agree from Theorem 2.
Note that F being Weierstrass, we have \({d}\le {\delta}\) and \({\delta}\log ({d})\in {{\mathcal {O}}}^{\sim}({\delta})\).
When F is Weierstrass, we have \({d}\le {\delta}\). Otherwise, we might have \({d}\notin {{\mathcal {O}}}^{\sim}({\delta})\).
We can extend this definition to non-Weierstrass polynomials, see Sect. 5.3.
We may allow \(m_1=B_1=0\) when considering non Weierstrass polynomials, see Sect. 5.3.
In [23, Prop.12], the condition \(P_k(0)\in \mathbb {K}_k^\times \) is imposed even if \(q_k=1\), but this has no impact from a complexity point of view.
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Poteaux, A., Weimann, M. Computing the equisingularity type of a pseudo-irreducible polynomial. AAECC 31, 435–460 (2020). https://doi.org/10.1007/s00200-020-00451-x
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DOI: https://doi.org/10.1007/s00200-020-00451-x