There are several questions about transverse complete intersection arising from L. Guth's paper:


<http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html>


We say a polynomial $P$ on $\mathbb{R}^n$ is nonsingular if for each point $x \in Z(P):=\{z\in \mathbb{R}^n\,|\, P(z)=0\}, $ we have that $ \nabla P(x) \neq 0.$


Suppose that $Q\_1, \cdots, Q\_k$ are polynomials on $\mathbb{R}^n$. We say that $Z(Q\_1,\cdots,Q\_k):=\{z\in \mathbb{R}^n\,|\, Q\_1(z)=0, \cdots, Q\_k(z)=0\}$ is a transverse complete intersection if for each point $x\in Z(Q\_1,\cdots,Q\_k), \nabla Q\_1(x), \cdots, \nabla Q\_k(x)$ are linearly independent.


The following are some claims which I did not figure out how to prove. Any idea would be appreciated.


(1). Suppose that $Q$ is a nonsingular polynomial on $\mathbb{R}^3$. For any non-zero vector $\omega$, we define $\mbox{Tan}\_\omega :=\{x \in Z(Q)\,|\, x+\omega \in T\_xZ(Q)\}=Z(Q,\nabla Q\cdot \omega).$ Then for almost every $\omega$, $\mbox{Tan}\_\omega=Z(Q,\nabla Q\cdot \omega)$ is a transverse complete intersection.


(2). Suppose that $Q$ is a nonsingular polynomial on $\mathbb{R}^n$, $\upsilon$ a fixed unit vector in $\mathbb{R}^n$. Define a smooth function $f: Z(Q)\longrightarrow \mathbb{R}$ by $f=\frac{(\nabla Q\, \cdot\, \upsilon )^2}{|\nabla Q|^2}$. Fix a point $x\_0 \in Z(Q)$. Prove that $\nabla Q(x\_0), \nabla f\, (x\_0)$ are linearly independent if and only if $\nabla f\, (x\_0) \neq 0.$


(3). Suppose that $Y=Z(Q\_1,Q\_2)$ is a transverse complete intersection in $\mathbb{R}^3$, $\upsilon$ a fixed unit vector in $\mathbb{R}^3$. Define a smooth function $f : Y \longrightarrow \mathbb{R}$ by $f=\frac{((\nabla Q\_1 \times \nabla Q\_2)\,\cdot \,\upsilon)^2}{|\nabla Q\_1 \times \nabla Q\_2|^2}$. Fix a point $y\_0 \in Y$. Prove that $\nabla Q\_1(y\_0), \nabla Q\_2(y\_0), \nabla f\, (y\_0)$ are linearly independent if and only if $\nabla f \, (y\_0)\neq 0.$



about transverse complete intersection
The key idea to prove these claims is Sard's theorem from differential topology. A good reference for this area is the book Differential Topology by Guillemin and Pollack.


Here is an outline of the proof of the first claim. The other two are similar.


Z(Q) is a manifold in R^3, and moreover grad Q is non-vanishing on Z(Q). Consider the function


g: Z(Q) x S^2 to R given by


g(x, w) = grad Q(x) dot w.


A point (x,w) is a critical point for g if and only if w is parallel to grad Q(x). We know that grad Q(x) is non-vanishing on Z(Q), and so at every critical point of g, g(x,w) is non-zero. Therefore g^{-1}(0) is a codimension 1 submanifold of Z(Q) x S^2. Let M = g^{-1}(0).


Next, consider the projection


pi: M to S^2, given by pi(x,w) = w. The map pi is a smooth map. Note that pi^{-1}(w) is exactly Tan\_w \times {w}. Moreover, we claim that if w is a regular value of pi, then Tan\_w is a transverse complete intersection. This claim is similar to the proof of the Transversality Theorem on pages 68-69 of Guillemin-Pollack. Finally, by Sard's theorem, almost every w in S^2 is a regular value of pi.


