Let $P(x,t)=t^d+a_{d-1}(x)t^{d-1}+\ldots +a_0(x)$ with coefficients that are $\mathbb{C}$-analytic in the polydisc $D=\left\{x\in\mathbb{C}^m : \max |x_j|<r \right\}$. Assume that there exists a continuous function $\gamma :D\rightarrow \mathbb{C}$ such that $P(t,\gamma(t))\equiv 0$.
Is $\gamma$ necessarily an analytic function?