Multiplicities#
This section contains some useful invariants for an isolated singularity at the origin of a hypersurface given by a multivariate polynomial \(f\).
- MilnorNumber(f): RngMPolElt -> RngElt#
Given a polynomial \(f\in K[x_1, \ldots, x_n]\), where \(K\) is a field, return the Milnor number of \(f\) at the origin. This is the dimension of the quotient by the ideal generated by the partials of \(f\) in the localization of \(K[x_1, \ldots, x_n]\) at the origin. See [Cox et al., 1998, p. 147] or [Decker and Lossen, 2006, Remark 9.37].
- TjurinaNumber(f): RngMPolElt -> RngElt#
Given a polynomial \(f\in K[x_1, \ldots, x_n]\), where \(K\) is a field, return the Tjurina number of \(f\) at the origin. This is the dimension of the quotient by the ideal generated by \(f\) and the partials of \(f\) in the localization of \(K[x_1, \ldots, x_n]\) at the origin. See [Cox et al., 1998, p. 148] or [Decker and Lossen, 2006, Def. 9.35].
- Example: Variety (ex-e4a04f)#
We compute some Milnor and Tjurina numbers, based on Exercise 12 of [Cox et al., 1998, p. 177].
> P<x,y> := PolynomialRing(RationalField(), 2); > MilnorNumber((x^2 + y^2)^3 - 4*x^2*y^2); // 4-leaved rose 13 > [MilnorNumber(y^2 - x^n): n in [1 .. 5]]; [ 0, 1, 2, 3, 4 ] > P<x,y,z> := PolynomialRing(RationalField(), 3); > [MilnorNumber(x*y*z + x^n + y^n + z^n): n in [1 .. 10]]; [ 0, 1, 8, 11, 14, 17, 20, 23, 26, 29 ] > [TjurinaNumber(x*y*z + x^n + y^n + z^n): n in [1 .. 10]]; [ 0, 1, 8, 10, 13, 16, 19, 22, 25, 28 ]
A much larger example is given in [Decker and Lossen, 2006, p. 254].
> P<x,y> := PolynomialRing(RationalField(), 2); > f := y^2 - 2*x^28*y - 4*x^21*y^17 + 4*x^14*y^33 - 8*x^7*y^49 + > x^56 + 20*y^65 + 4*x^49*y^16; > time TjurinaNumber(f); 2260 Time: 0.010