AI 中文总结
针对齐次多项式系统的结式系统构造问题，提出基于线性组合的新方法，得到更小基数的结式系统，提升了上界并构造了显式低基数系统。
AI 中文摘要
对于n个变量中s个d次齐次多项式构成的系统f=0，我们研究构造结式系统的问题。结式系统是输入多项式系数的有限多项式集合，其消失可表征具有公共非零解的系统f。经典构造结式系统的方法要么依赖大型系数矩阵的极大子式，要么依赖输入多项式的一般线性组合的结式系数，通常会产生包含大量多项式的结式系统。我们开发了基于输入多项式线性组合的结式构造的新方法，得到了基数较小的结式系统。主要结果为：1）证明存在具有${d+n-1 \brace n-1} s-n^2+1$个多项式的结式系统，每个多项式都是输入多项式的n个线性组合的结式，该结果优于已知的上界，即使对于双变量齐次多项式系统亦是如此；2）在输入多项式非零的假设下，当n固定时，构造了基数为poly(s,d)的显式结式系统。
英文摘要
For a system of $s$ homogeneous polynomials of degree $d$ in $n$ variables, say ${\bf{f}} = 0$, we consider the problem of constructing resultant systems. A resultant system is a finite set of polynomials in the coefficients of the input polynomials, the vanishing of which characterizes the systems $\bf{f}$ with a common non-zero solution. The classical approaches for constructing resultant systems rely either on maximal minors of large coefficient matrices or on the coefficients of a resultant of generic linear combinations of the input polynomials. Typically, they produce resultant systems containing a very large number of polynomials. We develop new constructions based on taking resultants of linear combinations of the input polynomials; this results in resultant systems of small cardinality. Our main results are: 1) We prove that a resultant system with ${d+n-1 \choose n-1} s-n^2+1$ polynomials exists; each polynomial is the resultant of $n$ linear combinations of the input polynomials. This improves the previously known upper bounds, even for systems of bivariate homogeneous polynomials. 2) Under the assumption that the input polynomials are non-zero, we construct explicit resultant systems with cardinality $\mathrm{poly}(s,d)$, when $n$ is fixed.
