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 $\nabla f \, (y_0)\neq 0.$