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I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not. One way I know is the cylindrical algebraic decomposition (CAD) algorithm. ...

Suppose I have a polynomial Morse function $f: \mathbb{R}^n \to \mathbb{R}$. Consider the ideal $I(\nabla f)$ generated by the partial derivatives $\partial_i f$, and assume that the real zero-set of ...