## Hilbert's Nullstellensatz is in the Polynomial Hierarchy (1996)

Venue: | Journal of Complexity |

Citations: | 39 - 9 self |

### BibTeX

@ARTICLE{Koiran96hilbert'snullstellensatz,

author = {Pascal Koiran},

title = {Hilbert's Nullstellensatz is in the Polynomial Hierarchy},

journal = {Journal of Complexity},

year = {1996},

volume = {12},

pages = {273--286}

}

### OpenURL

### Abstract

We show that if the Generalized Riemann Hypothesis is true, the problem of deciding whether a system of polynomial equations in several complex variables has a solution is in the second level of the polynomial hierarchy. In fact, this problem is in AM, the "Arthur-Merlin" class (recall that NP ` AM ` RP NP ` \Pi 2 ). The best previous bound was PSPACE. An earlier version of this paper was distributed as NeuroCOLT Technical Report 96-44. The present paper includes in particular a new lower bound for unsatisfiable systems, and remarks on the ArthurMerlin class. 1 A part of this work was done when the author was visiting DIMACS at Rutgers University. 1 Introduction In its weak form, Hilbert's Nullstellensatz states that a system f 1 (x) = 0; : : : ; f s (x) = 0 (1) of polynomial equations in n unknowns has no solution over C if and only if there are polynomials g 1 ; : : : ; g s 2 C [X 1 ; : : : ; X n ] such that P s i=1 f i g i = 1. For this reason, the problem of deciding whethe...

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