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Exponential-Polynomial Equations

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Exponential-Polynomial Equations M. Zoeteman Exponential-polynomial equations Master thesis July 30, 2019 Thesis supervisor: Dr. J.-H. Evertse Leiden University Mathematical Institute Contents 1 Introduction 3 1.1 Lower bounds . .4 1.2 Outline of the thesis . .5 2 Linear recurrence sequences and exponential-polynomial equations 5 2.1 Linear recurrence sequences . .5 2.2 General exponential-polynomial equations . .9 3 Linear equations 10 3.1 Results on linear equations . 10 3.2 Some ideas and techniques behind the proofs of Theorems 3.5 and 3.6 . 12 3.2.1 Algebraic number theory and heights . 12 3.2.2 The Subspace Theorem . 13 3.2.3 Sketch of the proofs . 14 4 Proof of Theorem 1.4 14 4.1 Two results from the paper of Corvaja, Schmidt and Zannier . 14 4.1.1 Proof of Theorem 4.1 . 15 4.1.2 Proof of Theorem 4.2 . 18 4.2 A specialisation argument . 24 4.3 Schmidt's paper . 24 4.3.1 Some reductions . 24 4.3.2 Reducing our main equation to a determinant equation . 25 4.3.3 A case distinction . 26 4.4 Application of Theorem 4.2 to equation (4.25) . 27 4.5 Proof of Theorem 1.4 by induction . 28 4.6 Proof of Corollary 1.5 . 29 2 1 Introduction Let α1; :::; αn 2 C be non-zero multiplicatively independent numbers. We write α = (α1; :::; αn), n and for x = (x1; :::; xn) 2 Z we use the notation x x1 xn α = α1 ··· αn : We consider the Diophantine equation x n α = f(x) in x 2 Z ; (1.1) where f(x) = f(x1; :::; xn) 2 C[x1; :::; xn] is a polynomial of total degree δ. By a special case of a theorem of Laurent [6, 7, 8] the above equation has only finitely many solutions. Explicit upper bounds for the number of solutions have been studied as well. Let K be the field n+δ K := Q(α1; :::; αn) and write ∆ := n and B := ∆ + 1. In 2000, Schlickewei and Schmidt [15] proved the following upper bound in the case where K is a number field. 2 3 Theorem 1.1. If K is a number field of degree d, then (1.1) has at most d6B 235B solutions. This is in fact a special case of what Schlickewei and Schmidt proved, namely Theorem 2.12 in x2, which gives an upper bound for the number of solutions to so called exponential-polynomial equations, of which (1.1) is a special case. Then, in 2009, Schmidt [19] generalised this result as follows. Theorem 1.2. If the set of all roots of unity in K generates a number field of finite degree d0, 6B2 9B then (1.1) has at most d0 exp B solutions. Notice that the previous two results only give an upper bound under certain assumptions on the αi, and that the upper bounds depend on the αi. Building further on Schmidt's ideas, Corvaja, Schmidt and Zannier [2] were in 2010 the first to prove an upper bound that only depends on the number of variables n and the total degree δ of f. Their result is as follows. Theorem 1.3. Equation (1.1) has at most exp B9B solutions. In this thesis we prove the following sharpening of the upper bound in Theorem 1.3. 6 Theorem 1.4. Equation (1.1) has at most (8B)9B solutions. Writing the upper bounds as 6 (8B)9B = exp 9B6 log(8B) and exp B9B = exp (exp (9B log B)) ; we see that our gain is in loosing one exponential. In [2], it is remarked that Theorem 1.3 is easily generalised to the same upper bound for the number of rational solutions to (1.1), if we fix xi xi log αi the values of log α1; :::; log αn (so that we can define αi := e for xi 2 Q). In the following corollary, we generalise Theorem 1.4 in this way. The proof is given in x4.6. Corollary 1.5. If we fix values for log α1; :::; log αn, then the equation x1 xn n α1 ··· αn = f(x1; :::; xn) in x 2 Q (1.2) 6 has at most (8B)9B solutions. 3 1.1 Lower bounds One could wonder whether the upper bound in Theorem 1.4 is close to being sharp as n ! 1. Obviously there is no general lower bound for the number of solutions to (1.1). For example, if the αi are algebraically independent and f 2 Q[x1; :::; xn] and f(0; :::; 0) 6= 1, then (1.1) has no solutions at all. The question remains whether there are particular classes of equations of the form (1.1) for which the number of solutions is large when n ! 1, and in particular whether this number can get close to the upper bound from Theorem 1.4. Therefore we consider classes of equations x1 xn α1 ··· αn = f(x1; :::; xn) for which the number of solutions g(n) satisfies g(n) ! 1 as n ! 1. We construct two of such examples. Example 1.6. For any multiplicatively independent complex numbers α1; :::; αn, let f be the n linear polynomial f(x1; :::; xn) := α1x1 + ::: + αnxn. Let ei 2 Z denote the vector with i-th coordinate equal to 1 and the other coordinates equal to 0. Then e1; :::; en are n solutions to n+1 (1.1) for our choice of f. For this f we have δ = 1, so B = n + 1 = n + 2, and (1.1) has at least n ≥ B − 2 solutions. f(n) Notation. For functions g; h : >0 ! >0 we use the notation f(n) ∼ g(n) if lim = 1. Z R n!1 g(n) p n n In the following example we apply Stirling's estimate n! ∼ 2πn e in order to study the growth of the number of solutions as a function of n. Example 1.7. Let again α1; :::; αn be any multiplicatively independent complex numbers. For n a vector y = (y1; :::; yn) 2 f0; 1g , we write jyj := y1 + ::: + yn. Now let f be the polynomial X jyj y1 yn f(x1; :::; xn) := (−1) α1 ··· αn (1 − y1 − x1) ··· (1 − yn − xn); y2f0;1gn x n 2n then we have f(x) = α for each x 2 f0; 1g . Since f has total degree n, we have ∆ = n , so by Stirling's estimate (2n)! 4n B ∼ ∆ ∼ ∼ p : (n!)2 πn p Thus, for n large enough we have B ≤ 4n, and the equation (1.1) has at least 2n ≥ B solutions. This is a smaller number of solutions than in Example 1.6, but in this new example the total degree δ of f satsfies δ ! 1 as n ! 1, while the polynomial in Example 1.6 is linear for any n. p There is still a large gap between the lower bounds B −2 and B that we found in our examples, 6 and the upper bound (8B)9B from Theorem 1.4. To the best of the author's knowledge, such lower bounds have not been studied yet in the literature, so it is an open challenge to "close the gap", i.e. to construct equations of the form (1.1) with more solutions and/or to improve the upper bound from Theorem 1.4. In the following example we fix n = 1, but the degree δ of f tends to infinity. Example 1.8. Consider the equation αx = f(x) in x 2 Z, where α 2 C is not a root of unity and where 0 1 k k X BY x − j C i f(x) = B C α ; i − j i=1 @j=1 A j6=i 4 with k a positive integer. Notice that for all integers 1 ≤ x ≤ k we have f(x) = αx. In this 1+k x example we have B = 1 + 1 = k + 2, and our equation α = f(x) has at least k = B − 2 solutions. 1.2 Outline of the thesis We now provide a sketch of how we improve the upper bound from Theorem 1.3 to the one from Theorem 1.4. Let K be a field of characteristic 0, let n ≥ 2 be an integer and let Γ ⊂ (K∗)n be ∗ a subgroup of rank r. For coefficients a1; :::; an 2 K , we consider the equation a1x1 + ::: + anxn = 1 in (x1; :::; xn) 2 Γ: (1.3) A solution to this equation is called non-degenerate if no proper subsum of the left-hand side vanishes. Evertse, Schlickewei and Schmidt [3] proved in 2002 that the number of non-degenerate solutions to (3.4) is at most exp (6n)3n(r + 1) . In 2009, Amoroso and Viada [1] improved this 4 upper bound to (8n)4n (n+r+1). In x4 we prove Theorem 1.4. In this proof we follow the argu- ments given in the proofs of Theorem 1.2 and Theorem 1.3. At the point where these proofs apply the upper bound of Evertse, Schlickewei and Schmidt, we apply the new upper bound from Amoroso and Viada. Using this new upper bound in the further calculations, we will arrive at our result as stated in Theorem 1.4. In x2 we study some general theory of exponential-polynomial equations and their relation to linear recurrence sequences. In x3, we discuss some basic terminology related to linear equations, the results of Evertse, Schlickewei and Schmidt, and of Amoroso and Viada, and (very briefly) some of the techniques behind the proofs of these results.
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