compute the genus of an algebraic curve
genus(f, x, y)
squarefree polynomial specifying an algebraic curve
The genus of an irreducible algebraic curve is a non-negative integer. It equals the dimension of the holomorphic differentials. It also equals (d-1)(d-2)/2 minus the sum of the delta invariants, which can be computed with algcurves[singularities]. Here d is the degree of the curve.
The polynomial f must be squarefree and have degree at least 1, otherwise an error message follows. A complete irreducibility check is not performed, only a few partial tests.
f ≔ x4+x2⁢y+y2
Warning, negative genus so the curve is reducible
f ≔ subs⁡z=1,761328152⁢x6⁢z4−5431439286⁢x2⁢y8+2494⁢x2⁢z8+228715574724⁢x6⁢y4+9127158539954⁢x10−15052058268⁢x6⁢y2⁢z2+3212722859346⁢x8⁢y2−134266087241⁢x8⁢z2−202172841⁢y8⁢z2−34263110700⁢x4⁢y6−6697080⁢y6⁢z4−2042158⁢x4⁢z6−201803238⁢y10+12024807786⁢x4⁢y4⁢z2−128361096⁢x4⁢y2⁢z4+506101284⁢x2⁢z2⁢y6+47970216⁢x2⁢z4⁢y4+660492⁢x2⁢z6⁢y2−z10−474⁢z8⁢y2−84366⁢z6⁢y4:
This f is a polynomial of degree 10 having a maximal number of cusps according to the Plucker formulas. It was found by Rob Koelman. It has 26 cusps and no other singularities, hence the genus is (10-1)*(10-2)/2 - 26 = 10.
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