A result of I.V. Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of degree at most three. In this note we describe homaloidal polynomials in three variables of arbitrarily large degree in positive characteristic. Using combinatorial arguments, we also classify line arrangements whose polar map is homaloidal in positive characteristic.

中文翻译：

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正特征任意大程度的平面极地克雷莫纳图

IV Dolgachev的结果表明，三个变量的复同形多项式，即极坐标图为双正态的复齐次多项式，最多为3级。在本说明中，我们以正特征中任意大程度的三个变量描述了同胚多项式。使用组合参数，我们还对极性特征为正性的极坐标图的线排列进行了分类。