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A.Skutin
A.Skutin
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Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$$f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\setminus\{0\}$$\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.

Is it true that $\mathbb{C}[x_1,\ldots, x_n] = \mathbb{C}[f_1,\ldots, f_n]+f_n\cdot\mathbb{C}[x_1,\ldots, x_n]$$\mathbb{C}[x, y] = \mathbb{C}[f, g]+g\cdot\mathbb{C}[x, y]$?

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\setminus\{0\}$.

Is it true that $\mathbb{C}[x_1,\ldots, x_n] = \mathbb{C}[f_1,\ldots, f_n]+f_n\cdot\mathbb{C}[x_1,\ldots, x_n]$?

Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.

Is it true that $\mathbb{C}[x, y] = \mathbb{C}[f, g]+g\cdot\mathbb{C}[x, y]$?

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A.Skutin
A.Skutin
  • 329
  • 2
  • 13

Question about Jacobian subalgebra

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\setminus\{0\}$.

Is it true that $\mathbb{C}[x_1,\ldots, x_n] = \mathbb{C}[f_1,\ldots, f_n]+f_n\cdot\mathbb{C}[x_1,\ldots, x_n]$?