In this paper we study polynomial vector fields on \({\mathbb {C}}^{2}\) of degree larger than 2 with \(n^{2}\) isolated singularities. More precisely, we show that if two polynomial vector fields share \(n^{2}-1\) singularities with the same spectra (trace and determinant) and from these singularities \(n^{2}-2\) have the same positions, then both vector fields have identical position and spectra at all the singularities. Moreover we also show that if two polynomial vector fields share \(n^{2}-1\) singularities with the same positions and from these singularities \(n^{2}-2\) have the same spectra, then both vector fields have identical position and spectra at all the singularities. Moreover we also prove that generic vector fields of degree \(n>2\) have no twins and that for any \(n>2\) there exist two uniparametric families of twin vector fields, i.e. two different families of vector fields having exactly the same singular points and for each singular point both vector fields have the same spectrum.

在本文中，我们研究了\（{\ mathbb {C}} ^ {2} \]大于2且具有\（n ^ {2} \）孤立奇点的多项式矢量场。更准确地说，我们表明，如果两个多项式矢量场共享具有相同光谱（迹线和行列式）的\（n ^ {2} -1 \）奇异性，并且从这些奇异性\（n ^ {2} -2 \）具有相同的位置，则两个矢量场在所有奇点处都具有相同的位置和光谱。此外，我们还表明，如果两个多项式向量字段共享具有相同位置的\（n ^ {2} -1 \）奇点，并且从这些奇点\（n ^ {2} -2 \）具有相同的光谱，则两个矢量场在所有奇点处都具有相同的位置和光谱。此外，我们还证明了度数为\（n> 2 \）的一般矢量场没有孪生，并且对于任何\（n> 2 \）都存在两个单参数的孪生矢量场族，即两个不同的向量场族具有完全相同的相同的奇异点，对于每个奇异点，两个矢量场具有相同的频谱。