Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ?
Thanks.
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1$\begingroup$ That is a hard problem. Suppose $f=P(X)Q(Y)R(Z)$, where $P,Q,R$ are univariate polynomials that are pairwise coprime. Finding the Euler characteristic boils down to finding the number of real roots of each of these polynomials. $\endgroup$– Liviu NicolaescuSep 18, 2015 at 20:18
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$\begingroup$ @LiviuNicolaescu Very interesting, could you just explain why the Euler characteristic boils down to finding the number of real roots of each of these polynomials? $\endgroup$– sphereSep 18, 2015 at 20:34
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$\begingroup$ I cannot fix the TeX typo in my comment above. Take for example the polynomial $$f=P(x)(y^2+1)(z^2+1).$$Then $V(f)$ is a union of planes, one plane for each real root of $P$. A formula for the Euler characteristic would lead to a formula forthe number of real roots. There are ways of determining this number, Hermite matrix, Sturm sequence, but for higher degrees they are not very practical. $\endgroup$– Liviu NicolaescuSep 19, 2015 at 11:39
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1 Answer
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A “formula” is a lot to ask for, but there are algorithms based on Morse theory.
E.g. §5 of Basu (1999), or §3 of Fortuna-Gianni-Luminati (2004).
answered Sep 18, 2015 at 20:13
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$\begingroup$ Wow! thank you for the references! I'm impressed by those papers. $\endgroup$– sphereSep 18, 2015 at 20:25