Suppose that a polynomial $p=\sum_{0\leq j<n} u_jx^j$ with coefficients $0\leq u_j<n$ is constantly 0 modulo some $n$. If $n$ is a prime number, then $u=0$, because the Lagrange polynomials of degree $n-1$ form a basis and the coefficients of $p$ in that basis evaluate to 0. If $u\neq0$, then contrapositively $n$ is decomposable. Is there an efficient technique for finding such $a$ and $b$, that $n=ab$?
By efficient, I mean that it should be applicable to RSA numbers $n$ such as $$\begin{align}\text{RSA}_{230} = &1796949159794106673291612844957324615636756180801260007088891883553172646\\&0341490933493372247868650755230855864199929221814436684722874052065257937\\&4956943483892631711525225256544109808191706117425097024407180103648316382\\&88518852689\end{align}$$ with a week of computing time on a personal computer, assuming we are given $p$ in a sparse format with a small number of nonzero coefficients $u_j\neq0$, say $10^6$.
One answer is if all coefficients are $0$, assuming this is not the case,
then it could happen just if the power $j$ and the power $i$ which differ by $\phi(n)$ which is Euler totient function have the same coefficient meaning that $p= \cdots + u_j *x^j+ \cdots +u_i *x^i$ and $j-i = \phi(n)$ then $u_i = \pm u_j \mod n$ so you look at the polynomial and see a periodicity that tells you what $\phi(n)$ is and then by Fermat's little theorem and GCD(greates common divisor) you can factor $n$, how you get this amazing polynomial $p$ is another matter.
read about the discrete logarithm problem (open problem) to grasp how hard it would be to find such polynomial.
Example : for $n=77$ the polynomial $p=3 x^{61}+74*x$ which yield $0 \mod 77$ for any input $x$, so by the difference of the powers we can compute that $\phi(77)=60$ and use any one of the hundreds of algorithms that when given $\phi(n)$ factorize $n$.
Note : two important thing i found $p$ for $n=77$ because i computed $\phi(77)$ first (i worked backward lol).
secondly $p$ could have $2$ or $4$ or $6$ and so on terms.
another example for $n=77$ is the polynomial $p= 5*x^{65}+3*x^{61}-5*x^5+74*x$