  Polynomial multiplication is a fundamental problem in symbolic computation.
There are efficient methods for the multiplication of two univariate
polynomials. However, there is rarely efficiently nontrivial method for the
multiplication of two multivariate polynomials. Therefore, we consider a new
multiplication mechanism that involves a) reversibly reducing multivariate
polynomials into univariate polynomials, b) calculating the product of the
derived univariate polynomials by the Toom-Cook or FFT algorithm, and c)
correctly recovering the product of multivariate polynomials from the product
of two univariate polynomials. This work focuses on step a), expecting the
degrees of the derived univariate polynomials to be as small as possible. We
propose iterative Kronecker substitution, where smaller substitution exponents
are selected instead of standard Kronecker substitution. We also apply the
Chinese remainder theorem to polynomial reduction and find its advantages in
some cases. Afterwards, we provide a hybrid reduction combining the advantages
of both reduction methods. Moreover, we compare these reduction methods in
terms of lower and upper bounds of the degree of the product of two derived
univariate polynomials, and their computational complexities. With randomly
generated multivariate polynomials, experiments show that the degree of the
product of two univariate polynomials derived from the hybrid reduction can be
reduced even to approximately 3% that resulting from the standard Kronecker
substitution, implying an efficient subsequent multiplication of two univariate
polynomials.
