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Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or$f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$.This existent result suggests a possible way to deal with general Booleanfunctions via its subfunctions of some restrictions.  As an application, we consider the degree lower bound of representingpolynomials over finite rings. Let $f\in \mathcal{F}_{n}^*$ and denote theexact representing degree over the ring $\mathbb{Z}_m$ (with the integer $m>2$)as $d_m(f)$. Let $m=\Pi_{i=1}^{r}p_i^{e_i}$, where $p_i$'s are distinct primes,and $r$ and $e_i$'s are positive integers. If $f$ is symmetric, then $m\cdotd_{p_1^{e_1}}(f)... d_{p_r^{e_r}}(f) > n$. If $f$ is non-symmetric, by thesecond moment method we prove almost always $m\cdot d_{p_1^{e_1}}(f)...d_{p_r^{e_r}}(f) > \lg{n}-1$. In particular, as $m=pq$ where $p$ and $q$ arearbitrary distinct primes, we have $d_p(f)d_q(f)=\Omega(n)$ for symmetric $f$and $d_p(f)d_q(f)=\Omega(\lg{n}-1)$ almost always for non-symmetric $f$. Henceany $n$-variate symmetric Boolean function can have exact representing degree$o(\sqrt{n})$ in at most one finite field, and for non-symmetric functions,with $o(\sqrt{\lg{n}})$-degree in at most one finite field.

On Restricting No-Junta Boolean Function and Degree Lower Bounds by Polynomial Method