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.

翻译：多项式乘法是符号计算中的基本问题。对于两个一元多项式的乘法已有高效方法，但两个多元多项式乘法的非平凡高效方法仍较为罕见。为此，本文提出一种新型乘法机制，具体包含以下步骤：a）将多元多项式可逆地约化为一元多项式；b）通过Toom-Cook算法或FFT算法计算派生一元多项式的乘积；c）从两个一元多项式的乘积中正确恢复多元多项式的乘积。本文重点研究步骤a），致力于使派生一元多项式的次数尽可能低。我们提出迭代Kronecker代换方法，该方法采用更小的代换指数替代标准Kronecker代换。同时，我们将中国剩余定理应用于多项式约化，并发现其在某些场景下的优势。随后，我们提出融合两种约化方法优势的混合约化方案。进一步地，我们从两个派生一元多项式乘积次数的上下界及其计算复杂度两个维度，对这些约化方法进行了比较分析。基于随机生成的多元多项式实验表明，经混合约化得到的两个一元多项式乘积次数，可降至标准Kronecker代换结果的约3%，这为后续高效完成两个一元多项式乘法提供了保证。