Algebraic Settings for the Problem P6=NP?"

Lectures in Applied Mathematics Volume 00, 19xx Algebraic Settings for the Problem \P 6= NP?" Lenore Blum, Felip e Cucker, MikeShub, and Steve Smale Abstract. When complexity theory is studied over an arbitrary unordered eld K , the classical theory is recaptured with K = Z . The fundamental 2 result that the Hilb ert Nullstellensatz as a decision problem is NP-complete over K allows us to reformulate and investigate complexity questions within an algebraic framework and to develop transfer principles for complexity theory. Here we show that over algebraically closed elds K of characteristic 0 the fundamental problem \P 6= NP?" has a single answer that dep ends on the tractability of the Hilb ert Nullstellensatz over the complex numb ers C .Akey comp onent of the pro of is the Witness Theorem enabling the eliminationof transcendental constants in p olynomial time. 1. Statement of Main Theorems We consider the Hilb ert Nullstellensatz in the form HN=K : given a nite set of p olynomials in n variables over a eld K , decide if there is a common zero over K . At rst the eld is taken as the complex numb er eld C . Relationships with other elds and with problems in numb er theory will b e develop ed here. This article is essentially Chapter 6 of our b o ok Complexity and Real Compu- tation to b e published by Springer. Background material can b e found in [Blum, Shub, and Smale 1989]. Only machines and algorithms which branchon\hx = 0?" are considered here. The symbol is not used. Thus the development is quite algebraic, eventually using prop erties of the height function of algebraic numb er theory. A main theme is eliminating constants. The moral is roughly: using transcendental and algebraic numb ers do esn't help much in sp eeding up integer decision problems. Let Q b e the algebraic closure of the rational numb er eld Q. The following will b e proved. Theorem 1. If P=NP over C , then P= NP over Q , and the converse is also true. 1991 Mathematics Subject Classi cation. 68Q Computer Science, Theory of Computation, 11G Numb er Theory, Arithmetic Algebraic Geometry. Blum was partially supp orted by the Letts-Villard Chair at Mills College. Cucker was partially supp orted by DGICyT PB 920498, the ESPRIT BRA Program of the EC under contracts no. 7141 and 8556, pro jects ALCOM I I and NeuroCOLT. Cucker, Smale, and Shub were partially supp orted by NSF grants. c 0000 American Mathematical So ciety 0075-8485/00 $1.00 + $.25 p er page 1 2 LENORE BLUM, FELIPE CUCKER, MIKE SHUB, AND STEVE SMALE Remark 1. Here C may b e replaced byany algebraically closed eld contain- ing Q . Nowwe are going to de ne an invariant of integers and p olynomials over Z which describ es how many arithmetic op erations are necessary to build up an integer starting from 1. More precisely,acomputation of length l of the integer m is a sequence of integers, x ;x ;::: ;x where x =1,x =mand given k , 0 1 l 0 l 1 k l , there are i; j ,0i; j < k such that x = x x where is addition, k i j subtraction or multiplication. We de ne : Z ! N by m is the minimum length of a computation of m. The following is easy to check, where here and in the sequel log denotes log . 2 Proposition 1. For al l m 2 N one has m 2 log m. k 2 If m is of the form 2 , then m = log log m + 1. The same is essentially true even if m is anypower of 2. Op en Problem. Is there a constant c such that c k ! log k all k 2 N? We remark that if \factoring is hard" using inequailities then the op en problem 1 has a negative answer. Definition 1. Given a sequence of integers a wesay that a is easy to com- k k c pute if there is a constant c such that a log k , all k>2, and hard to k compute otherwise. Wesay that the sequence a is ultimately easy to compute if k there are non-zero integers m such that m a is easy to compute and ultimately k k k hardtocompute otherwise. In Sections 5 and 6 we will prove: Theorem 2. If the sequence of integers k ! is ultimately hardtocompute, then HN=C , the Hilbert Nul lstel lensatz over C , is intractable and hence P 6=NP over C . Thus in that case, P 6=NP over Q . Next consider the analogous situation for p olynomials with integer co ecients f 2 Z[t]. A computation of length l of f is a sequence of u 2 Z[t] where u =1, i 0 u = t, u = f and given k ,1k lthere are i; j ,0i; j < k such that 1 l u = u u where is addition, subtraction or multiplication. De ne : Z[t] ! N k i j by f is the minimum length of a computation of f . Let Zerf b e the numb er of distinct integral zeros of f . The following has a certain plausibility: c Hypothesis. Zerf f for all non-zero f 2 Z[t]. Here c is a universal constant. We don't know if the hyp othesis is true or false, even, for example, with the constant c =1. 1 Here is a sketch of the pro of. Supp ose to the contrary that k !iseasy to compute and n is the pro duct of primes p and q where p<k <q.We will showhow to easily factor n. c Let x ;x ;:::;x = k! b e a short computation of k !, l log k . Then we induce a short 0 1 l computation of r = k !modnusing x mo d n; x mo d n;:::;r = x mo d n: 0 1 l By the Euclidean Algorithm, y = gcdr;nmay b e easily computed. By our hyp othesis it follows that y = p, and thus our assertion is proved. ALGEBRAIC SETTINGS FOR THE PROBLEM \P 6= NP?" 3 Theorem 3. If the above Hypothesis is true then NP 6=P over C , and NP 6=P over Q . 2. Eliminating constants: Easy cases In this section we b egin a study of the problem of eliminating the constants of a computation without an exp onential increase in the time. Our rst result asserts that this can always b e done if the constants lie in an algebraic extension of the given eld K . The result holds for elds of anycharacteristic. To start we brie y and informally recall some notation and de nitions. Supp ose M is a machine over a commutative ring or eld R with unit. Then b oth the 1 1 input and output spaces of M are R , and the state space is R . Here R is the 1 disjoint union 1 n R = [ R n0 and R is the bi-in nite direct sum space over R. Elements of R have the form 1 1 x =::: ;x ;x ;x :x ;x ;::: 2 1 0 1 2 where x 2 R for all integers i, x = 0 for jk j suciently large, and : is a dis- i k tinguished marker b etween x and x . We call x the \i-th co ordinate" of x and 0 1 i x ;::: ;x the \ rst n co ordinates" of x. For brevity, and when the intentis 1 n clear from context, we sometimes omit the negative co ordinates and write elements of the state space as n; x ;x ;::: ;x ;0;::: orx;x ;::: ;x ;0;::: , or even 1 2 n 1 2 n x ;x ;::: ;x . 1 2 n The machine's input map, asso ciated with its input node, takes a p oint x 2 n 1 R R and maps it to ::: ;0;0;n : x ;x ;::: ;x ;0;0;::: 2 R : 1 2 n 1 Here n is the size of x. The output map, asso ciated with the machines's output node, takes a p oint x =::: ;m : x ;x ;::: 2 R and maps it to x ;::: ;x if m is a p ositive 1 2 1 1 m 0 integer, to the unique p ointofR if m = 0, and is unde ned otherwise. These maps make sense if the characteristic of R is 0. If the characteristic is p ositive, we replace n and m here by the appropriate numb er of 1's to the left of the distinguished marker. Machines also have computation, shift and branch nodes with asso ciated oper- ations p olynomial or rational maps, right/left shifts and the identity and asso ciated next state and next node maps. Without loss of generality,we assume that at branch no des, machines branch right or left dep ending on whether or not the rst co ordinate x of the current state x is 0. 1 1 A decision problem over R is a pair Y; Y where Y Y R . Here Y 0 0 is the set of problem instances and Y is the set of yes instances. For example, 0 HN=K is a decision problem over K where Y = f nite p olynomial systems over K g and Y = f nite p olynomial systems over K that are solvable over K g. A nite 0 1 p olynomial system over K is represented as an elementof K assuming some standard listing of its co ecients.

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