凸半代数集上有理点的计算与SOS分解
Computing rational points in convex semi-algebraic sets and SOS decompositions
摘要：
设${\cal P}=\{h_1, ..., h_s\}\subset \Z[Y_1, ..., Y_k]$、$D\geq °(h_i)$为$1\leq i \leq s$、$σ$为$h_i$的系数的位长度的边界，以及$Φ$为定义凸半代数集的无量词${\cal P}$公式。我们设计了一个算法，当且仅当${\cal S}\cap \Q\neq\emptyset$返回一个合理的点在${\cal S}$。它需要$σ^{\bigO(1)}D^{\bigO(k^3)}$位操作。如果输出有理点，则其坐标具有由$σD^{\bigO(k^3)}$支配的位长度。利用这一结果，我们得到一个程序，决定如果一个多项式$f\in \Z[X_1, >..., X_n]$是一个平方和的多项式在$\Q[X_1, ..., X_n]$。用$d$表示$f$的程度，$τ$表示$f$、$D={{n+d}\choose{n}}$和$k\leq D(D+1)-{{n+2d}\choose{n}}$中的系数的最大比特长度。该过程需要$τ^{\bigO(1)}D^{\bigO(k^3)}$位操作，并且输出的多项式的系数具有由$τD^{\bigO(k^3)}$支配的位长度。
英文摘要：
Let ${\cal P}=\{h_1, ..., h_s\}\subset \Z[Y_1, ..., Y_k]$, $D\geq °(h_i)$ for $1\leq i \leq s$, $σ$ bounding the bit length of the coefficients of the $h_i$'s, and $Φ$ be a quantifier-free ${\cal P}$-formula defining a convex semi-algebraic set. We design an algorithm returning a rational point in ${\cal S}$ if and only if ${\cal S}\cap \Q\neq\emptyset$. It requires $σ^{\bigO(1)}D^{\bigO(k^3)}$ bit operations. If a rational point is outputted its coordinates have bit length dominated by $σD^{\bigO(k^3)}$. Using this result, we obtain a procedure deciding if a polynomial $f\in \Z[X_1, >..., X_n]$ is a sum of squares of polynomials in $\Q[X_1, ..., X_n]$. Denote by $d$ the degree of $f$, $τ$ the maximum bit length of the coefficients in $f$, $D={{n+d}\choose{n}}$ and $k\leq D(D+1)-{{n+2d}\choose{n}}$. This procedure requires $τ^{\bigO(1)}D^{\bigO(k^3)}$ bit operations and the coefficients of the outputted polynomials have bit length dominated by $τD^{\bigO(k^3)}$.
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