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 [CLO98, p. 147] or [DL06, 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 [CLO98, p. 148] or [DL06, Def. 9.35].

Example: Milnor and Tjurina Numbers (ex-e4a04f)

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We compute some Milnor and Tjurina numbers, based on Exercise 12 of [CLO98, 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 [DL06, 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

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