## Experimental Mathematics

- Experiment. Math.
- Volume 13, Issue 4 (2004), 441-449.

### Testing the Logarithmic Comparison Theorem for Free Divisors

F. J. Castro-Jiménez and J. M. Ucha-Enríquez

#### Abstract

We propose in this work a computational criterion to test if a free divisor {\small $D\subset {\bf C}^n$} verifies the *Logarithmic Comparison Theorem* (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of {\small $D$} in {\small ${\bf C}^n$}.

For Spencer free divisors {\small $D\equiv(f=0)$}, we solve a conjecture about the generators of the annihilating ideal of {\small $1/f$} and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.

#### Article information

**Source**

Experiment. Math., Volume 13, Issue 4 (2004), 441-449.

**Dates**

First available in Project Euclid: 22 February 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1109106436

**Mathematical Reviews number (MathSciNet)**

MR2118269

**Zentralblatt MATH identifier**

1071.14024

**Subjects**

Primary: 14F50

Secondary: 32C38: Sheaves of differential operators and their modules, D-modules [See also 14F10, 16S32, 35A27, 58J15] 32C35: Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30] 13Pxx: Computational aspects and applications [See also 14Qxx, 68W30] 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]

**Keywords**

de Rham cohomology Logarithmic Comparison Theorem free divisors Gröbner bases

#### Citation

Castro-Jiménez, F. J.; Ucha-Enríquez, J. M. Testing the Logarithmic Comparison Theorem for Free Divisors. Experiment. Math. 13 (2004), no. 4, 441--449. https://projecteuclid.org/euclid.em/1109106436