Whenever, people talk about singular plane curves they talk about their Newton polytope which is obviously coordinate dependent. I understand that with some conditions over the singular curve, some invariantes can be calculated from the Newton polytope e.g the multiplier ideal of the monomial ideal $(x^p,y^q)$ is the same than the one of the polynomial $x^p+y^q$. I am also aware that under some non degeneracy conditions you can use the Newton polytope for finding Hodge numbers, Milnor number, etc. However, I am not able to find a reference for equisingularity i.e topological equivalent. Two different curves with the same Newton polytope are equisingular? If not, can I impose restriction over the curves for having this property?
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The answer is yes.
More precisely, if two nondegenerate Newton singularities $f(x,y)=0$ and $g(x,y)=0$ have the same Newton polygon, then they are topologically equivalent.
For a reference, look at Takamura's book "Splitting deformations of degenerations of complex curves III", pag. 138, in particular at the note after Theorem 7.4.1.
answered Sep 9, 2011 at 12:53
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3$\begingroup$ Note that the word "non-degenerate" is important here. For example, $x^2+3xy+y^2+y^3$ and $x^2+2xy+y^2+y^3$ have the same Newton polygon, but the former is a node and the latter is a cusp. That is because the latter is degenerate -- see Takamura's book for more. $\endgroup$ Sep 9, 2011 at 13:07
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$\begingroup$ @David: good point (and nice example) $\endgroup$ Sep 9, 2011 at 22:05
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$\begingroup$ Thanks both of you, the reference is precise and the example eloquent. $\endgroup$– mathStSep 12, 2011 at 17:56