Counterexamples to witness conjectures. Notations Let E be the set of admissible constant expressions built up from Z; + ;?;;/; exp and log. Here a co

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1 Counterexamples to witness conjectures Joris van der Hoeven D pt. de Math matiques (b t. 45) Universit Paris-Sud Orsay CEDEX France August 15, 003 Consider the class of exp-log constants, which is constructed from the integers using the eld operations, exponentiation and logarithm. Let z be such an exp-log constant and let n be its size as an expression. Witness conjectures attempt to give bounds $(n) for the number of decimal digits which need to be evaluated in order to test whether z equals zero. For this purpose, it is convenient to assume that exponentials are only applied to arguments with absolute values bounded by 1. In that context, several witness conjectures have appeared in the literature and the strongest one states that it is possible to choose $(n) = O(n). In this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures. Keywords: witness conjecture, counterexample, zero testing, elementary constant, high precision fraud A.M.S. subject classification: 11Y60, 68W30, Introduction Consider the class of exp-log constant expressions, which is constructed from the integers using the eld operations, exponentiation and logarithm. An important problem in computer algebra is to test whether an exp-log constant expression c represents zero. A straightforward approach is to evaluate c up to a certain number of decimal digits and test whether this evaluation vanishes. Witness conjectures attempt to give bounds $(n) for the number of decimal digits which are necessary as a function of the size n of the expression c. Of course, exponentials can be used in order to produce massive cancellations, like in e ee10 +e?ee10? e ee10? 1 0: For this reason, it is appropriate to allow only for exp-log expressions such that jsj61 for all subexpressions of the form e s. In that context, several witness conjectures appeared in the literature [vdh97, vdh01a, vdh01b, Ric01], and the strongest one states that we may take $(n) = O(n). In this paper we give a counterexample to this strong witness conjecture. The counterexample is based on the observation that it suces to nd a counterexample for the power series analogue of the problem [vdh01b] and a suggestion made by D. Richardson. In section 4, we will generalize our technique and give counterexamples to all witness conjectures with $(n) = n O(1). However, for this generalization, we need to extend the notion of exp-log constants so as to include algebraic numbers. 1

2 Counterexamples to witness conjectures. Notations Let E be the set of admissible constant expressions built up from Z; + ;?;;/; exp and log. Here a constant is said to be admissible if it evaluates to a real number. Given c E, we denote by (c) N its size (the number of inner roots in the corresponding expression tree plus the number of digits which are needed to write the integers at the leafs) and by c R its evaluation. We denote by C E the subset of all expressions c, such that jsj 6 1 for all subexpressions of the form e s. Consider the ring R = Q[[z]] of formal power series. Let S be the set of series expressions built up from z, elements in Q, the ring operations and left composition of expressions which represent innitesimal series by one of the series I = z/(1? z) L = log(1 + z) E = exp(z)? 1 Given such an expression f S, we denote by (f ) N its size (the number of nodes of the corresponding expression tree) and by f R the represented series. We also denote by # z f the number of occurrences of z in f and by v(f ) the valuation of f. Given f S and g S with v(g) > 0, the substitution of g for z in f yields another series expression f g in S and we have (f g) = (f ) + (# z f ) ((g)? 1) ; (1) v(f g ) = v(f ) v(g): () Similarly, given f S and c C, such that jcj is suciently small, the substitution of c for z in f yields a constant expression f (c) C of size (f (c)) = ~(f ) + (# z f ) ((c)? 1); (3) where ~(f ) is the number of inner nodes of f plus the sizes of the rational numbers on the leafs. For c! 0, we also have log jf (c)j v(f ) log jcj: Proposition 1. Given f S with v(f ) > 0 and k N, we have (f k ) = ((f )? # z f ) (# z f ) k? 1 # z f? 1 + (# z f ) k ; (4) v(f k ) = v(f ) k : (5) If c C is such that jcj is suciently small, then we also have (f k (c)) = (~(f )? # z f ) (# z f ) k? 1 # z f? 1 + (# z f ) k (c): (6) Proof. This follows from (1), () and (3) by a straightforward induction. 3. The strong witness conjecture Consider = log(1? log(1? z/))? z S:

3 Joris van der Hoeven 3 We have () = 11, # z = and v( ) = 3, since = 1 4 z3 + O(z 4 ): Theorem. Let $ be a witness function with $(n) = O(n ) and < log 3/log. Then there exists an expression S of size n with 0 and v( ) > $(n). Proof. By proposition 1, we have n 3 ( k ) = 10 k? 9 k and v( k ) = 3 k. It therefore suces to take = k for a suciently large k. Theorem 3. Let $ be a witness function with $(n) = O(n ) and < log 3/log. Then there exists a constant expression c C of size n with c 0 and jcj 6 e?$(n). Proof. On the interval [0; 1 ], we notice that satises 06 (c)6c 3. Hence, j k ( 1 )j6?3k for all k. By proposition, we also have n 3 ( k ( 1 ))k for large k. Therefore, it suces to take c = k ( 1 ) for a suciently large k. 4. Polynomial witness conjectures Let E^, C^ and S^ be the analogues of E, C and S, if we replace Z and Q by the set of algebraic numbers Q^ in their respective denitions. The size of an algebraic number c is dened to be the minimal size of a polynomial equation satised by c. After choosing a suitable determination of log, the evaluations of constants in E^ are complex numbers. The analogues of all observations in section remain valid. Given l > and a = (a 0 ; ; a l ) (Q^ ) l+1, we denote a = (a 0 z) log(1 + z) (a 1 z) (a l?1 z) log(1 + z) (a l z) S^: Lemma 4. Given a; b (Q^ ) l+1 with b a, we have b a. Proof. Let i be maximal such that b i a i. Modulo postcomposition of both sides of the equation b = a with [log(1 + z) (a i+1 z) log(1 + z) (a l z)]?1 ; we may assume without loss of generality that i = l. Then b admits a singularity above z =?b?1 l, near to which b log l?1 (z + b l ). On the other hand, the number of nested?1 logarithms in the logarithmic transseries expansion of a near any point above z =?b l cannot exceed l? 1. Therefore, we must have b a. Lemma 5. There exist a; b (Q^ ) l+1 with = b? a 0 and v() > l. Proof. The mapping from (Q^ ) l+1 into Q^ l, which maps a to the rst l Taylor coecients of a, is polynomial. Since dim (Q^ ) l+1 > dim Q^ l, this mapping cannot be injective. We conclude by the previous lemma. Theorem 6. Let $ be a witness function with $(n) = O(n log n ) < log? log 3 4 Then there exists an expression S^ of size n with 0 and v( ) > $(n).

4 4 Counterexamples to witness conjectures Proof. With as in lemma 5, consider = k for large l N and l m log k = log l :? log Since () = 6 l + 7, v( ) > l and # z =, proposition implies n 3 and From (7) it follows that log n = Plugging this into (8), we obtain () = (6 l + 9) ( k? 1) + k l k l log?log (7) v( ) > l k?log = e log l+o(log l) : (8)? log log l + O(1): v( ) > e log n+o(log n) ; which clearly implies the theorem, by choosing l large enough. Theorem 7. Let $ be a witness function with $(n) = O(n ) and R >. Then there exists an expression c C^ of size n with c 0 and jcj 6 e?$(n). Proof. With as in lemma 5, choose l such that log l/log >. Then for r Q > \ [0; 1 ] suciently small, the closed disk B r = fz C: jzj 6 rg is mapped into itself and j (z)j6z l for z B r. Now proposition implies n 3 ( k (r)) k and j k (r)j 6 r lk for large k. Therefore, c = k (r) yields the desired counterexample for a suciently large k. 5. Algebraic counterexamples The technique from the previous section may also be used in order to produce algebraic counterexamples. Indeed, given l > 0 and a = (a 0 ; ; a l ) (Q^ ) l+1, let p p 1 + z (a 1 z) (a l?1 z) 1 + z (a l z) a = (a 0 z) Then we have the following analogue of lemma 4: Lemma 8. Given a; b (Q^ ) l+1 with b a, we have b a. Proof. Consider the Riemann surface of a admits an algebraic singularity at z = z l =? 1 a l of degree. On one of the two leafs, we again have an algebraic singularity at z = z l?1 =? 1 a l + 1 a l a l?1 of degree, and so on for the z l? ; ; z 1 given by z z i = (z? 1) z z (z? 1) z a l a l?1 a i+1 a i (?1): We conclude by the observation that the mapping (a 1 ; ; a l ) (z 1 ; ; z l ) is injective.

5 Joris van der Hoeven 5 6. Conclusion We have given counterexamples to the most optimistic kind of witness conjectures. In the power series context, we previously proved a witness conjecture for a doubly exponential witness function $ [SvdH01]. Hopefully, this technique may be extended in order to yield Khovanskii-like bounds $(n) = e O(n ) [Kho91]. This leaves us with the question what happens for growth rates between e O(log n) and e O(n ). In particular, it would be very useful for practical applications if the witness conjecture would hold for a witness function of exponentiality 0 (i.e. log k $ exp k Id for suciently large k). It might also be interesting to further investigate the proof technique used in this paper. For instance, can we do without algebraic numbers? Would it be possible to replace by a? z in lemma 5? Can we make theorem 7 as strong as theorem 6? Does there exist an approximation theory for power series by expressions of the form a (analogous to Pad approximation)? Bibliography [BB9] J.M. Borwein and P.B. Borwein. Strange series and high precision fraud. Mathematical Monthly, 99:6640, 199. [Kho91] A. G. Khovanskii. Fewnomials. American Mathematical Society, Providence, RI, [Ric94] D. Richardson. How to recognise zero. J. Symbol. Comput., 4(6):67646, [Ric01] D. Richardson. The uniformity conjecture. In Lecture Notes in Computer Science, volume 064, pages 537. Springer Verlag, 001. [SvdH01] J.R. Shackell and J. van der Hoeven. Complexity bounds for zero-test algorithms. Technical Report , Pr publications d'orsay, 001. [vdh97] J. van der Hoeven. Automatic asymptotics. PhD thesis, cole polytechnique, France, [vdh01a] J. van der Hoeven. Fast evaluation of holonomic functions near and in singularities. JSC, 31:717743, 001. [vdh01b] J. van der Hoeven. Zero-testing, witness conjectures and dierential diophantine approximation. Technical Report 001-6, Pr publications d'orsay, 001.

2 Section 2 However, in order to apply the above idea, we will need to allow non standard intervals ('; ) in the proof. More precisely, ' and may gene

2 Section 2 However, in order to apply the above idea, we will need to allow non standard intervals ('; ) in the proof. More precisely, ' and may gene Introduction 1 A dierential intermediate value theorem by Joris van der Hoeven D pt. de Math matiques (B t. 425) Universit Paris-Sud 91405 Orsay Cedex France June 2000 Abstract Let T be the eld of grid-based