Towards Randomized Testing of $q$-Monomials in Multivariate Polynomials

Given any fixed integer $q\ge 2$, a $q$-monomial is of the format ${\displaystyle x_{i_1}^{s_1}{x}_{i_2}^{s_2}\mathrm{.}\mathrm{.}\mathrm{.}{x}_{i_t}^{s_t}}$ such that $1\le s_j \le q-1$,$1\le j \le t$\mathrm{.}$q$-monomials are natural generalizations of multilinear monomials. Recent research on testing multilinear monomials and $q$-monomails for prime $q$ in multivariate polynomials relies on the property that $Z_q$ is a field when $q\ge 2$ is prime. When $q>2$ is not prime, it remains open whether the problem of testing $q$-monomials can be solved in some compatible complexity. In this paper, we present a randomized $O^{\ast }({7.15}^k)$ algorithm for testing $q$-monomials of degree $k$ that are found in a multivariate polynomial that is represented by a tree-like circuit with a polynomial size, thus giving a positive, affirming answer to the above question. Our algorithm works regardless of the primality of $q$ and improves upon the time complexity of the previously known algorithm for testing $q$-monomials for prime $q>7$.