Algebraic curves and foliations

Consider a field k ${\mathsf {k}}$ of characteristic 0 $\hskip.001pt 0$ , not necessarily algebraically closed, and fixed algebraic curve f = $f=0$ defined by tame polynomial ∈ [ x y ] $f\in {\mathsf {k}}[x,y]$ with only quasi-homogeneous singularities. We prove that the space holomorphic foliations in plane A 2 $\mathbb {A}^2_{\mathsf having as invariant is generated -module at most four elements, three them are trivial d $fdx,fdy$ $df$ . Our proof algorithmic constructs fourth foliation explicitly. Using Serre's GAGA Quillen–Suslin theorem, we show for suitable extension K {K}}$ such module over {K}}[x,y]$ actually two therefore, curves free divisors sense K. Saito. After performing Groebner basis this module, observe many well-known examples, {K}}={\mathsf