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How to Recognize Zero 1 Introduction

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How to Recognize Zero 1 Introduction How to Recognize Zero Daniel Richardson Department of Mathematics University of Bath email dsrmathsbathacuk January Abstract An elementary p oint is a p oint in complex n space which is an isolated non singular solution of n equations in n variables each equation b eing either or the form p where p is a p olynomial in Qx x or of 1 n x i the form x e An elementary numb er is the p olynomial image j of an elementary p oint In this article a semi algorithm is given to decide whether or not a given elementary numb er is zero It is proved that this semi algorithm is an algorithm i e that it always terminates unless it is given a problem containing a counter example to Schanuels conjecture Intro duction In computing a lazy sequence is a nite initial segment of a sequence together with a pro cess which generates more elements of the sequence if desired See for example Paulson By analogy with this we may say that a lazy complex or real numb er is a b ounded precision oating p oint complex or real numb er together with a pro cess which could b e used to increase the precision to any desired extent We will say that two such lazy numb ers are equal if rep eated application of their pro cesses results in sequences which converge to the same ordinary numb er Of course there are other reasonable denitions of equality for lazy numb ers The one given here might b e called standard equality It will b e assumed in the following that real and complex numb ers are given in lazy form rather than as completed innities of some kind The following fundamental question immediately presents itself For which natural subsets of the real and complex numbers can we do exact computations The computations of interest include the eld op erations a test for equality among complex numb ers and determination of the sign of a real numb er It is clear that we can patch our approximations and pro cesses together in order to eect addition subtraction and multiplication If two numb ers are unequal we will eventually b e able to recognize this by calculating them to sucient precision However there may b e problems recognizing equality If we could recognize zero we could recognize equality since we can do subtraction If we can recognize zero we can also avoid mistakes with division by zero so we can do division Also if we can recognize zero we can order real numb ers eectively So the central part of the ab ove problem reduces to In which natural subsets of the real and complex numb ers can we recognize zero In the following section a denition is given for a subset of the complex numb ers which is called elementary This set will b e denoted by E E is algebraically closed and is also closed under application of elementary x functions such as e sinx cosx Also isolated solutions of systems of equations involving elementary functions and p olynomials with co ecients in E have co ordinates which are in E The main result b elow is that we can recognize zero among the elementary numb ers unless we are given a problem which contains a counterexample to Schanuels conjecture The Schanuel conjecture is explained b elow Let Q b e the rational numb ers If B is a set of complex numb ers and z is complex we will say that z is algebraically dep endent on B if there is a p olynomial d pt a t a 0 d in QB t with a d and pz 0 If S is a set of complex numb ers a transcendence basis for S is a subset B so that no numb er in B is algebraically dep endent on the rest of B and so that every numb er in S is algebraically dep endent on B The transcendence rank of a set S of complex numb ers is the cardinality of a transcendence basis B for S It can b e shown that all transcendence bases for S have the same cardinality Schanuels conjecture If z z are complex numb ers which are linearly 1 n z z n 1 g has transcendence rank at least e indep endent over Q then fz z e 1 n n It is generally b elieved that this conjecture is true but that it would b e extremely hard to prove The history of the zero recognition problem is somewhat confused by the fact that many p eople do not recognize it as a problem at all In the algebraic case the nature of the problem dep ends up on what we decide to accept as the denition of a complex algebraic numb er In general our way of understanding the algebraic numb ers has b een in uenced by the historical struggle to separate out the abstract algebra from interpretation in the complex numb ers From this p oint of view it has b een as sumed that algebra should avoid oating p oint approximations So it has b een considered that the right way to do an algebraic computation is to put all the numb ers involved into an algebraic numb er eld an abstract ob ject in which there is a canonical form See Frohlich and Shepherdson Note that we do not have a useful canonical form for the whole set of algebraic numb ers but only for the numb ers in each particular nitely generated algebraic numb er eld If we are ultimately interested in oating p oint numb ers it is not clear that it is sensible to construct an enclosing algebraic numb er eld in order to do one computation But in any case this option disapp ears when we work with elementary numb ers There is at present no suciently develop ed theory of elementary numb er eld We do not for example know which abstract elds with exp onentiation can b e emb edded into the complex numb ers The ideal of separation b etween algebra and geometric interpretation do es not seem to work very well in this case The rst go o d result ab out recognition of zero among non algebraic numb ers is due to Caviness Caviness shows in essence that if a weak version of the Schanuel conjecture is true it is p ossible to dene a canonical form and thus to solve the zero recognition problem in the set of numb ers which are obtained by starting with the rationals and i and closing under addition subtraction multiplication and exp onentiation Of course this set is probably not algebraically closed More recently Wilkie and Macintyre have proved that if the Schanuel con x jecture is true then the theory of R e is decidable where R is the ordered eld of the reals In particular the Wilkie and Macintyre result solves the zero recognition problem for the necessarily elementary numb ers in the minimal mo del for this x theory Their metho ds can b e extended also to the theory of R e sin x [01] where sin x means sinx restricted to the interval and dened to [01] b e outside this interval In this theory all the real elementary numb ers are denable The real and imaginary part of complex elementary numb ers are real elementary So the metho ds of Wilkie and Macintyre can b e used to show that the zero recognition problem can b e solved for the elementary numb ers The work rep orted in this article is the result of a long indep endent devel opment however the intention of which is ultimately to develop algorithms to solve problems in the real and complex numb ers See Richardson The techniques used here ie Wus metho d and the LLL algorithm have their origins in computer algebra rather than in mo del theory Elementary p oints and numb ers Denition An exponential system in variables x x is S E where 1 n r k S p p is a list of r polynomials in Qx x and E w r 1 r 1 n k 1 z z z i k 1 with fw w z z g is a list of k terms w e w e e 1 k 1 k i k fx x g 1 n Let C b e the complex numb ers In all the following we will use S E to denote an exp onential system r k as describ ed ab ove We will use J S E to denote the r k by n matrix of r k partial derivatives f x where f f S and f f E i j 1 r r r +1 r +k k n Denition If is a point in C we wil l say that S E is non singular at r k if r k n and if the matrix J S E is non singular at r k n Denition A point in C is elementary if there is an exponential system S E with r k n so that S E and so that S E is non r k r k r k singular at Denition A complex number c is elementary if there is an elementary point and a polynomial q in Qx x so that c q 1 n At each stage in the following we will assume that we have approximated pr n some elementary numb ers to within some tolerance We will use pr n for the numb er of decimal places which are currently assumed to b e known n If x x is in C dene dx x to b e the maximum of the absolute 1 n 1 n values of the real and imaginary parts of x x ie 1 n dx x M axj Rex j j I mx j j Rex j j I mx j 1 n 1 1 n n n For in C let N f d g 2n N can b e visualized as a co ordinate aligned b ox in R around We will use N to denote the b oundary of N It is assumed that we have an interval arithmetic pro cedure for p olynomials x x n 1 with the following prop erty if p is an expression for e in x x e 1 n such a p olynomial and is an n tuple of pr n precision complex oats then the interval arithmetic pro cedure applied to p over N gives a pair of intervals I I with rational endp oints so that the b ox which is the pro duct of them r i in the complex plane is guaranteed to contain the image of N under p Furthermore the pro cedure is such that the lengths of I and I tend to zero as r i pr n increases See Alefeld and Herzb erger We will say p in N if the intervals I I pro duced by the pr n r i interval arithmetic pro cedure do not b oth contain zero ie if the complex is not in the b ox which is the pro duct of the intervals We will say possibl ep N in the complementary case in which the intervals I I pro duced by the interval arithmetic
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