Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let ${\mathbb{K}}_C〚x_1,\cdots ,x_e〛$ be the ring of formal power series in several variables with exponents in a line free cone C. We consider irreducible polynomials $f=y^n+a_1(\underset{\_}{x})y^{n-1}+\cdots +a_n(\underset{\_}{x})$ in ${\mathbb{K}}_C〚x_1,\cdots ,x_e〛[y]$ whose roots are in ${\mathbb{K}}_C〚x_1^{\frac{1}{n}},\cdots ,x_e^{\frac{1}{n}}〛$. We generalize to these polynomials the theory of Abhyankar-Moh. In particular we associate with any such polynomial its set of characteristic exponents and its semigroup of values. We also prove that the set of values can be obtained using the set of approximate roots. We finally prove that polynomials of $\mathbb{K}〚x_1,\cdots ,x_e〛[y]$ fit in the above set for a specific line free cone (see Section 4).

让 $\mathbb{ķ}$ 是特征为零的代数封闭场，令 ${\mathbb{ķ}}_C〚X_{\mathrm{1个}}，\cdots ，X_{Ë}〛$是线性自由锥C中有指数的几个变量的形式幂级数的环。我们考虑不可约的多项式$F={ÿ}^{ñ}+{\mathrm{一种}}_{\mathrm{1个}}（\underset{\_}{X}）{ÿ}^{ñ-\mathrm{1个}}+\cdots +{\mathrm{一种}}_{ñ}（\underset{\_}{X}）$ 在 ${\mathbb{ķ}}_C〚X_{\mathrm{1个}}，\cdots ，X_{Ë}〛[ÿ]$ 根源于 ${\mathbb{ķ}}_C〚X_{\mathrm{1个}}^{\frac{\mathrm{1个}}{ñ}}，\cdots ，X_{Ë}^{\frac{\mathrm{1个}}{ñ}}〛$。我们将这些理论推广到Abhyankar-Moh理论。特别是，我们将其特征指数集和值的半群与任何此类多项式相关联。我们还证明了可以使用一组近似根获得一组值。我们最终证明的多项式$\mathbb{ķ}〚X_{\mathrm{1个}}，\cdots ，X_{Ë}〛[ÿ]$ 在上面的设置中适合特定的自由圆锥体（请参见第4节）。