In the book ‘Computational Complexity’ by Arora and Barak the following question is posed (exercise 2.20.):
Let REALQUADEQ be the language of all satisfiable sets of quadratic equations over real variables. Show that REALQUADEQ is NP-complete.
I know how to show NP-hardness, but I’m stuck when it comes to proving that this problem is in NP, in particular how to show that we can describe a solution using a polynomial number of bits.
I did some research and found out that over the complex numbers, it remains an open question if the problem is in NP [1]. It also seems closely related to the existential theory of the reals, which again is not known to be in NP.
Thus my question: Is this problem known to be in NP? And if so, could somebody point me in the right direction regarding the proof.
[1] Pascal Koiran, Hilbert’s Nullstellensatz is in the Polynomial Hierarchy, Journal of Complexity 12 (1996), no. 4, pp. 273–286.
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