A semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities having real coefficients and is a union of finitely many maximally connected components. We consider the problem of deciding whether two given points in a semi-algebraic set are connected; that is, whether the two points lie in the same connected component. In particular, we consider the semi-algebraic set defined by f <> 0 where f is a given polynomial with integer coefficients. The motivation comes from the observation that many important or non-trivial problems in science and engineering can be often reduced to that of connectivity. Due to its importance, there has been intense research effort on the problem. We will describe a symbolic-numeric method based on gradient ascent. The method will be described in two papers. The first paper (the present one) will describe the symbolic part and the forthcoming second paper will describe the numeric part. In the present paper, we give proofs of correctness and termination for the symbolic part and illustrate the efficacy of the method using several non-trivial examples.

中文翻译：

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半代数集合中的连通性 I

半代数集是由多项式方程和具有实系数的不等式定义的实空间子集，并且是有限多个最大连通分量的并集。我们考虑决定半代数集中的两个给定点是否连通的问题；即两点是否在同一个连通分量中。特别地，我们考虑由 f <> 0 定义的半代数集，其中 f 是具有整数系数的给定多项式。动机来自观察到科学和工程中许多重要或非平凡的问题通常可以简化为连通性问题。由于其重要性，对该问题进行了大量的研究工作。我们将描述一种基于梯度上升的符号数字方法。该方法将在两篇论文中描述。第一篇论文（现在的一篇）将描述符号部分，即将发表的第二篇论文将描述数字部分。在本文中，我们给出了符号部分的正确性和终止证明，并使用几个非平凡的例子说明了该方法的有效性。