Geometry theorem proving using Hilbert's Nullstellensatz

The theory of elementary algebra and elementary geometry was shown to be decidable by Tarski using a quantifier elimination technique in the 1930’s [26]. Subsquently, Tarski’s decision algorithm was improved by others notably among them Seidenberg [25], Monk [23], and Collins [12], and recently by Ben-Or et al [4]. These methods are algebraic and are based on translating geometry statements into first-order formulae using the operations 0, 1, -1, +, *, 2, = of an ordered field with variables rangmg over real numbers. Among these decision procedures, Collins’s method based on cylinderical algebraic decomposition technique is, to our knowledge, the only decision procedure implemented so far; see [2, 31 for details.