Interior derivative estimates and Bernstein theorem for Hessian quotient equations

In this paper, we obtain the interior derivative estimates of solutions for elliptic and parabolic Hessian quotient equations. Then we establish the Bernstein theorem for parabolic Hessian quotient equations, that is, any parabolically convex solution $u=u(x,t)\in C^{4,2}({\mathbb{R}}^n\times (-\mathrm{\infty },0])$ for $-u_t\frac{S_n(D^{2}u)}{S_l(D^{2}u)}=1$ in ${\mathbb{R}}^n\times (-\mathrm{\infty },0]$ must be the form of $u=-mt+P(x)$ with $m>0$ being a constant and $P$ being a convex quadratic polynomial.