Abstract
We give formulas for the image Milnor number of a weighted-homogeneous map-germ \(({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^{n+1},0)\), for \(n=4\) and 5, in terms of weights and degrees. Our expressions are obtained by a purely interpolative method, applied to a result by Ohmoto. We use our approach to recover the formulas for \(n=2\) and 3 due to Mond and Ohmoto, respectively. For \(n\ge 6\), the method is valid as long as certain multi-singularity conjecture holds.
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Notes
There seems to be a typo in Sharland’s parameterisation of \({{\hat{N}}}_1\). Our term \(x^4y\) replaces her \(x^2y\), inconsistent with the claim that \({{\hat{N}}}_1\) unfolds \({{\hat{E}}}_1\).
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The authors were partially supported by the ERCEA 615655 NMST Consolidator Grant and by the Basque Government through the BERC 2014-2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.
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Pallarés, I., Peñafort Sanchis, G. Image Milnor Number Formulas for Weighted-Homogeneous Map-Germs. Results Math 76, 152 (2021). https://doi.org/10.1007/s00025-021-01418-1
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DOI: https://doi.org/10.1007/s00025-021-01418-1