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UserX2017
UserX2017
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Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ($0\in L$) a complex line and $H=F(L)$ such that $g|_H\not=0$$g^{-1}(0)\cap H=\{0\}$. It is true that $h=g|_H\circ F|_L$ has topological degree less than or equal to k? This is true if F is linear map!

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ($0\in L$) a complex line and $H=F(L)$ such that $g|_H\not=0$. It is true that $h=g|_H\circ F|_L$ has topological degree less than or equal to k? This is true if F is linear map!

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ($0\in L$) a complex line and $H=F(L)$ such that $g^{-1}(0)\cap H=\{0\}$. It is true that $h=g|_H\circ F|_L$ has topological degree less than or equal to k? This is true if F is linear map!

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UserX2017
UserX2017
  • 81
  • 7

Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ($0\in L$) a complex line and $H=F(L)$ such that $g|_H\not=0$. It is true that $h=g|_H\circ F|_L$ has topological degree less than or equal to k? This is true if F is linear map!