We describe conditions on the signed support, that is, on the set of the exponent vectors and on the signs of the coefficients, of a multivariate polynomial $f$ ensuring that the semi-algebraic set $\{ f < 0 \}$ defined in the positive orthant has at most one connected component. These results generalize Descartes' rule of signs in the sense that they provide a bound which is independent of the values of the coefficients and the degree of the polynomial. Based on how the exponent vectors lie on the faces of the Newton polytope, we give a recursive algorithm that verifies a sufficient condition for the set $\{ f < 0 \}$ to have one connected component. We apply the algorithm to reaction networks in order to prove that the parameter region of multistationarity of a ubiquitous network comprising phosphorylation cycles is connected.

翻译：我们描述了多元多项式$f$的符号支撑集（即指数向量集合及系数符号）所需满足的条件，这些条件可确保正象限中定义的半代数集$\{ f < 0 \}$至多只有一个连通分支。这些结果推广了笛卡尔符号法则，其推论给出一个与系数值及多项式次数无关的界。基于指数向量在牛顿多胞体面上的分布方式，我们提出了一种递归算法，用于验证$\{ f < 0 \}$具有单个连通分支的充分条件。我们将该算法应用于反应网络，以证明包含磷酸化循环的普遍网络中多稳态参数区域是连通的。