The BCSS model of computation  extends the classical computational complexity theory to arbitrary fields/rings, in particular to , , and , posing the conjectures , , and . Here the machine is assumed to work with equality comparisons over , and inequality comparisons over and . In the case of , the bit cost model is assumed. The purpose of this paper is to show that the theory presented in the first paper of the series  naturally extends to these cases, answering the open questions raised in . We first note that the separation of P and NP/poly, as proved in , already implies (see the introduction of  for relevant references). In this paper we make the separation over explicit, and also settle the case for and :
The problem of determining the feasibility of a set of quadratic equations (over , , and ) with variables requires at least operations in the BCSS model of computation.
We denote the underlying field/ring by , , or . For and , the scheme representing the computational problem of interest will be defined over the algebraic closure of , which is . We consider the problem QUAD whose instances are polynomial systems over consisting of quadratic equations. The equations of a given instance are assumed to have a common solution in , where is the number of variables. QUAD is NP-complete over and as proved in , which also shows NP-completeness over with the extra requirement that the norm of any point in the solution set is bounded.
We will define the amplifying functor from QUAD to the category of schemes over . It is a subfunctor of the Hilbert functor, which is representable by a projective scheme over by the following result.
Theorem 2 ().
Let be a projective scheme over . Then for every polynomial , there exists a projective scheme over , which represents the functor . Furthermore, the Hilbert functor is represented by the Hilbert scheme
We consider . In this case, the connectedness of the Hilbert scheme for a fixed Hilbert polynomial was established by Hartshorne . Note first that given a homogenized polynomial , one might consider the closed subscheme
so that each polynomial equation and hence a polynomial system identifies a closed subscheme of via the corresponding ideal. In particular, we refer to the Hilbert polynomial of an instance.
Given an instance of QUAD, a non-empty subset of the equations of the corresponding polynomial system is said to form a sub-instance of QUAD. As in an instance, a sub-instance also has a corresponding polynomial system induced by the polynomial equations it contains. Two sub-instances are called disjoint if both do not belong to a single instance of QUAD. A sub-problem of QUAD is a computational problem induced by a subset of the disjoint sub-instances of QUAD. A sub-problem of QUAD is called a simple sub-problem if the polynomial systems of have the same solution set and the same Hilbert polynomial. Given two sub-instances and of defined via the same variable set , is said to be a variant of if it can be obtained from by replacing each in a subset of by in its equations for some , followed by a permutation of the variables. Two sub-instances of QUAD are said to be distinct if
they are not variants of each other,
the polynomial systems that define them correspond to distinct ideals,
so that distinct sub-instances of correspond to distinct points in moduli. A simple sub-problem of is said to be homogeneous if the following two conditions hold.
The instances of are pair-wise distinct.
None of the instances of can be expressed as the Cartesian product of sets of sub-instances of defined via disjoint sets of variables, for some .
We now give a partial definition of the amplifying functor, which we denote by . Let be a homogeneous simple sub-problem of QUAD consisting of a set of polynomial systems defined via the variables . Let be the homogenized -th polynomial in the polynomial system . Define
We define the product of computational problems in the usual sense, i.e. the set of instances of the product is the Cartesian product of the sets of instances of the problems. Given a sub-problem of , for some , where are homogeneous simple sub-problems of defined via disjoint sets of variables, we set
By the connectedness of the Hilbert scheme for a fixed Hilbert polynomial, we have that there is a functor , which is an extension of , such that a homogeneous simple sub-problem always maps to a connected scheme via . Thus, we extend to , overriding the definition of the amplifying functor.
We define to be the minimum number of deterministic operations required to solve . Given a homogeneous simple sub-problem of QUAD, we denote the number of instances of by . A sub-problem is called a normal sub-problem if , for some , where are homogeneous simple sub-problems defined via disjoint sets of variables. Over all such normal sub-problems , let denote the maximum value of . The proof of the following result is omitted, as the only change from the Fundamental Lemma of  is the underlying field/ring. The proof is oblivious to the method of comparison used by the machine (equality or inequality). It essentially uses the fact that a sub-problem with a single instance has non-zero complexity, which obviously holds for any type of machine.
3 Proof of Theorem 1
We inductively construct a homogeneous simple sub-problem of QUAD with instances, each having variables and equations, for . The result follows by Lemma 3 and the definition of .
For , consider first the instance with the following equations:
The first equation implies that at least one of and is , so that by the second and the third equations. Given these and the fourth equation, we have the following solution set: . Note that it has integer coordinates and bounded norm, a property that will be extended to the general case. The following are the equations of another instance.
By a similar argument, it has the solution set . Thus, both of the instances have the same constant Hilbert polynomial , resulting in a homogeneous simple sub-problem with instances. Assume now the induction hypothesis for some . In the inductive step, we introduce new variables , and new blocks of equations on these variables each consisting of equations in the exact form of the two instances given above. Appending these equations to each of the instances of the induction hypothesis, we obtain instances, which form a simple sub-problem. We make into a homogeneous simple sub-problem as follows. Given an instance, consider the graph whose nodes are the equations of the instance, and there is an edge between two nodes if they share a common variable. We impose that this graph be connected for all the instances, which is clearly satisfied in the base case . For the inductive step, replace the positive literal , where it appears in the fourth equation of its block, by the positive literal . This operation does not change the solution set and hence the Hilbert polynomial, as is already forced to be for all by the other equations. It also ensures connectivity of the graph across all the blocks including the newly introduced one, thus constructing a homogeneous simple sub-problem, completing the induction and the proof.
We have the following by Theorem 1 and the NP-completeness of QUAD over , , and .
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