  Given any fixed integer $q\ge 2$, a $q$-monomial is of the format
$\displaystyle x^{s_1}_{i_1}x^{s_2}_{i_2}...x_{i_t}^{s_t}$ such that $1\le s_j
\le q-1$, $1\le j \le t$. $q$-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 $O^*(7.15^k)$
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 $q>7$.

Computational Complexity