A Stothers–Mason theorem with a difference radical

Differential calculus is not a unique way to observe polynomial equations such as a+b=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a+b=c$$\end{document}. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Results, for example, on the non-existence of polynomial solutions to difference Fermat and difference Super-Fermat functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider these functional equations with non-polynomial entire solutions. Equations with polynomial or non-polynomial solutions are observed to see the sharpness of results obtained.


Introduction
The Stothers-Mason theorem states that if relatively prime polynomials a, b and c, not all of them identically zero, satisfy a + b = c, then deg c ≤ deg rad(abc) − 1, where the radical rad(abc) is the product of distinct linear factors of abc [27,37], see also [13,35]. An elementary application of Mason's theorem is that if x, y and z are non-trivial relatively prime polynomials satisfying x n + y n = z n , (1.1) where n ∈ N, then n ≤ 2. The Stothers-Mason theorem is a counterpart of the abc conjecture in number theory, while its consequence described above is Fermat's last theorem for polynomials (see, e.g., [24,25]). Fermat type functional equations, such as (1.1) and its generalizations, have been studied over many function fields [10,17,21] (see also, e.g., [15] and the references therein). For instance, if f n 1 + f n 2 + · · · + f n m = 1 ( 1 . 2 ) has a solution consisting of m polynomials f 1 , f 2 , . . . , f m , then n ≤ m 2 −m −1. For rational, entire and meromorphic solutions the corresponding bounds are n ≤ m 2 − 2, n ≤ m 2 − m and n ≤ m 2 − 1, respectively [17]. Hayman [18] calls the problem of finding the smallest m = G 0 (n) for which a solution of (1.2) exists as the Super-Fermat problem. A difference analogue of (1.2) was studied by the third author [26], who obtained similar bounds for a difference counterpart of (1.2) under certain conditions on the value distribution of solutions. The purpose of this paper is to introduce a difference counterpart of the radical, and to use it to prove a difference analogue of the Stothers-Mason theorem, as well as a truncated version of the difference second main theorem for holomorphic curves. As applications we prove results on the non-existence of polynomial or non-polynomial entire solutions to difference Super-Fermat functional equations.

Difference radical
Let p ≡ 0 be a polynomial in C[z], and let κ ∈ C\{0}. We define the κ-difference radical rãd κ ( p) of p as where d κ (w) = ord w ( p) − min{ord w ( p), ord w+κ ( p)} with ord w ( p) ≥ 0 being the order of zero of the polynomial p at w ∈ C. This corresponds to the way to define the usual radical rad p as where, denoting by p the derivative of p, d(w) = ord w ( p) − min ord w ( p), ord w ( p ) ∈ {0, 1}. Now, by definingñ κ ( p) = deg rãd κ ( p), it follows thatñ κ ( p) is the number of zeros of p appearing non-periodically with respect to the constant κ, where the multiplicities of the zeros are taken into account. In other words, n κ ( p) = w∈C (ord w ( p) − min{ord w ( p), ord w+κ ( p)}). (2.1) For example, if p has zeros of order 2, 1 and 3 at z 0 , z 0 + 1 and z 0 + 2, respectively, and no zero at z 0 + 3, then the zero of p at z 0 is counted once inñ 1 ( p) and the zero at z 0 + 2 three times inñ 1 ( p), while the zero at z 0 + 1 is not counted inñ 1 ( p).
In addition, we define κ p = p(z + κ) − p(z), and use the notation gcd( p, q) to denote the greatest common divisor of p and q over C[z]. Proof We may write p in the form where γ ∈ C and l i ∈ N ∪ {0}. Note that the roots of p are repeated in (2.2) the number of times according to their multiplicity, so the case β i = β k , i = k, is allowed. More precisely, for a zero of p(z), if ord β ( p) > ord β+κ ( p), then β is entered ord β ( p) − ord β+κ ( p) times as one of the '{β i }' in (2.2). If ord β ( p) ≤ ord β+κ ( p), then β is not entered as one of the '{β i }' in (2.2). Moreover, we may assume in (2.2) that β s = β t − (l t + 1)κ for any s, t = 1, 2, . . . , m, since otherwise, we can combine two products as Now, by (2.2), the difference radical satisfies the simple representation In fact, for each i ∈ {1, . . . , m} we have From (2.2), we have and so it follows that Therefore, Remark By Lemma 2.1, we have which corresponds to the well known expression for the classical radical, .
In what follows we denoten( p) = deg rad p for the number of all the distinct roots of p(z).

Difference analogue of the Stothers-Mason theorem
The Stothers-Mason theorem has been generalized in many different directions, for instance, to sums in one-dimensional function fields by Mason [28], by Voloch [43] and by Brownawell and Masser [2], to sums of pairwise relatively prime polynomials of several variables by Shapiro and Sparer [32], to sums in higher-dimensional function fields by Hsia and Wang [19], and to quantum deformations of polynomials by Vaserstein [41]. Motivated by the analogy between Diophantine approximation and Nevanlinna theory [42], the abc theorem has also been proven for complex entire functions by Van Frankenhuysen [39,40], for p-adic entire functions by Hu and Yang [20], and for non-Archimedean entire functions of several variables by Cherry and Toropu [5]. The following theorem is a difference analogue of the Stothers-Mason theorem, or in other words, a difference abc theorem for polynomials.
Proof Without loss of generality we may assume that max{deg a, deg b, deg c} = deg c. and and so gcd(a, κ a), gcd(b, κ b) and gcd(c, κ c) are all factors of a κ b − b κ a. Since a, b and c are relatively prime, it follows that also gcd(a, κ a), gcd(b, κ b) and gcd(c, κ c) are relatively prime. Therefore, then a κ b = b κ a, and so a is a factor of b κ a. Since a and b have no common factors, it follows that a is a factor of κ a. This is only possible if κ a ≡ 0. Similarly, under the assumption (3.6) it follows that κ b ≡ 0 and κ c ≡ 0, which contradicts the assumption of the theorem. Hence, (3.6) cannot hold and (3.5) is valid. By adding deg c to both sides of (3.5) and reorganizing the terms, we have The assertion follows by Lemma 2.1.
Example 3. 2 We can see that the assertion of Theorem 3.1 is sharp by the example a(z) where α, β ∈ C such that β = α = β ± κ. Namely, then a, b and c are relatively prime polynomials in C[z] such that a + b = c, and such that none of the differences κ a, κ b and κ c is identically zero. In addition, max{deg a, deg b, deg c} = 2, n κ (a) = 1, n κ (b) = 1, n κ (c) = 1 and n κ (a) + n κ (b) + n κ (c) − 1 = 2.
Comparing the statements of the Stothers-Mason theorem and Theorem 3.1 raises the question of whether a stronger statement would hold in Theorem 3.1. The following coprime polynomial identity is extremal to the Stothers-Mason theorem but not to Theorem 3.1. Nevertheless, this proves the conclusion of Theorem 3.1 cannot be (3.7).
Of course, when we choose the step size κ as −2, each of the latter three radicals has the same degree as its input. On the other hand, one observes that other known identities such as 4 do not work like the one that we chose here.
Then a, b and c satisfy (3.1), and max{deg a, The so-called Davenport inequality [7] for polynomials f (z), g(z) ∈ C[z] with f (z) 3 ≡ g(z) 2 (in answer to a question of Birch et al. [1]) is to which Stothers [37] and independently Zannier [45] gave the extremal functions for each degree. The simplest example is For example, instead of (2, 3)-pair due to Stothers, Shioda [33] proposed to call a triple { f , g, h} of polynomials a Davenport-Stothers triple (or a DS-triple) of order m if it satisfies the following condition: Then he continues: In this terminology, Stothers has proven among others the existence of DS-triples of order m for every m ≥ 1 and the finiteness of the number of essentially distinct DS-triples of order m. In fact, Davenport's elegant proof uses only linear algebra, while Stothers has generalized the Davenport inequality to the Stothers-Mason theorem along a geometric way to be mentioned shortly below. It should be mentioned here that Zannier gave Acknowledgement of priority [46] concerning his paper [45] that he recently discovered the paper [37] by Stothers, which actually covers part of his results, using a method of the same nature. This is almost a déjà vu for us and perhaps we should call this result the Davenport-Stothers-Zannier theorem in this note. By applying Theorem 3.1 we give . At this moment we cannot however find any other difference analogues to DS-triples, unfortunately. This is because the method by Stothers and others is in a geometric way based on the Riemann-Hurwitz relation which gives crucial information for the case of equality as Shioda mentioned in [34]. In fact, as well as in the case of equality for the Stothers-Mason theorem, they consider a Belyi map of the Riemann sphere P(C) which is rational and unramified outside {0, 1, ∞}. Such a strong method is not known yet in our difference setting, while we could find the above example through an ansatz by analogy to the simplest DS-triple as above.
In order to generalize Theorem 3.1 for m + 1 polynomials we recall the definition of the Casorati determinant, and introduce the corresponding difference radical.
Let a j (z) (1 ≤ j ≤ m) be linearly independent functions over the field of periodic functions of period κ ∈ C\{0}. Then the does not vanish identically. Many of the properties of Casoratians are similar to those of Wronskians, in part due to the correspondence of .
Note that when q = 1 these definitions reduce to the ones given in Sect. 2, that is, we have rãd [1] where |L i ∩(L i +1)∩· · ·∩(L i +q)| denotes the number of elements in the intersection of the set L i = {0, 1, . . . , l i } and its translations L i +k := {k, 1+k, . . . , l i +k} for k = 1, 2, . . . , q. This corresponds nicely to the usual radical rad q ( p) of truncation level q given by .
In fact, Hu and Yang [21] use the notation r q ( p) for the degree of rad q ( p) of truncation level q given by where α 1 , . . . , α l are distinct, m j ∈ N ( j = 1, . . . , l) and c = 0. By the above expression of The following theorem extends Theorem 3.1 for m + 1 polynomials, see, for example, [8,21,32] for the differential analogue. In the statement of the theorem we have chosen to follow the form used by Shapiro and Sparer [32], instead of the one due to de Bondt [8] with weaker assumptions. Extending the theorem using de Bondt's approach is beyond the scope of our paper, and it is left as an open question. Theorem 3.5 Let a 1 , . . . , a m+1 be pairwise relatively prime polynomials in C[z] such that a 1 + · · · + a m = a m+1 , (3.8) and such that a 1 , . . . , a m are linearly independent over C. Then, where we denotẽ and κ ∈ C\{0}.
Proof Now we consider the has also a zero at z = z 0 of multiplicity not smaller than Therefore, under the assumption of the pairwise relative primeness we see that by means of the notation (3.10).
On the other hand, the degree of C κ (z) is never beyond any sum of distinct m of the This implies our desired estimate By definitioñ Example 3.6 below shows that (3.11) does not hold in general when m > 2. If one wished to use the radicals rãd κ (a i ) in (3.9), it is possible to use such estimates as which can be sharp when m = 2 but the following example shows this is a crude estimate for our purposes.

Example 3.6
Given c ∈ C\{0}, we have the identity Thus we have a solution (a 1 , a 2 , a 3 , a 4 ) to the equation with m = 3, such that {a 1 , a 2 , a 3 } is a linear independent system of pairwise relatively prime polynomials by requiring c = 0, ±1, ± √ −1. In fact, we put and a 4 (z) = 2c 4 . Then we have max 1≤i≤4 {deg a i } = 4 and by observing the zeros of a i (i = 1, 2, 3), respectively. Hence it is not possible for us to On the other hand, this example gives so that this is far from an example to confirm whether our estimate is sharp, unfortunately. For this purpose, one needs to consider such an example that the a i (z) are of the form is positive at a zero of a i (z). Note that this quantity is always zero when n < m and the zeros of p i (z) appear non-periodically with respect to κ.
The following example observes the acuity of Theorem 3.5 in the case when m = 3 with κ = 1.

Example 3.7 Define
By simple computations, we see that a 1 (z) + a 2 (z) + a 3 (z) reduces to a polynomial, say a 4 (z), of degree at most 1 when α = −1/4 and Indeed, we have with (3.12) We can choose α so that a j (z), j = 1, 2, 3, 4 are pairwise relatively prime. Then a j satisfy (3.8), and max 1≤ j≤4 {deg a j } = 4,ñ [ 1 (a 4 ) = 1 which gives 4 ≤ 6, which is not enough to show the sharpness of Theorem 3.5 for the case m = 3. Next, we set α = i , and a 4 (z) reduces to a constant − 9 16 , which gives a somewhat sharper estimate 4 ≤ 5 for Theorem 3.5 when a m+1 is a constant.

Polynomial solutions of difference Super-Fermat functional equations
A factorial polynomial is defined as t n = t(t + 1) · · · (t + n − 1).
We extend this notation for the factorial of a polynomial p in C[z] as where the shift κ ∈ C\{0}.
As a consequence of Theorem 3.1 we obtain the following result on the non-existence of polynomial solutions to a difference Fermat functional equation. Proof Suppose first that none of a, b and c is constant. If (4.1) holds, then by Theorem 3.1, we have By repeating the same argument for b and c instead of a, we have and By combining (4.2), (4.3) and (4.4), it follows that and so n ≤ 2.
Assume now that at least one of a, b and c is constant. Then by (4.1) exactly one of them, say c, is constant. Then, by (4.2) and (4.3), we have which implies that n ≤ 1.
The following example shows that the assertion of Theorem 4.1 is sharp.
it follows that is not identically a constant. Such a complex structure with p > 0 encourages us to restrict our coefficient field to be of characteristic zero, even if the above singularity can be avoided by a choice κ = p/2 of the shift of κ .
The following theorem extends Theorem 4.1 to equations with arbitrarily many terms. for some κ ∈ C\{0} and n ∈ N, then Proof We may assume, without loss of generality, that [ p 1 ]n κ , [ p 2 ]n κ , . . . , [ p m ]n κ are linearly independent. For otherwise we may eliminate some of the polynomials from (4.5) to obtain a shorter equation, which is of the same form, but contains only linearly independent terms. Assume that m > n. Since for all m ≥ 2 the right hand side of (4.6) is always at least m, the inequality itself follows. Now we suppose that n ≥ m. By using Theorem 3.5 we obtain Therefore (4.7) gives which implies the assertion in the case n ≥ m.

Example 4.4
We consider the sharpness of the inequality (4.6) in the case m = 2. Let us first look at the case where the maximal degree of the polynomial solutions of (4.5) is one. In this case it can be seen by a direct substitution of arbitrary linear polynomials into (4.5) that such solutions are never relatively prime when n = 2. If the maximal degree of the polynomial solutions is two, then by Theorem 4.3 we have n ≤ 5/2. In Example 4.2 we have given a solution for the Eq. (4.5) with m = 2 and n = 2, which is optimal in this case, since n is an integer.  Choosing m = 2 in Corollary 4.5 implies the first assertion of Theorem 4.1, namely that n ≤ 2.
The final two results of this section deal with another canonical form of difference Fermat functional equations. for some κ ∈ C\{0} and n ∈ N, then Proof From Theorem 3.5 we have and so a similar discussion as in the proof of Theorem 4.3 implies the assertion. If at least one of the polynomials in the Eq. (4.5) is constant, then (4.5) reduces into (4.8). In particular, when m = 2, Corollary 4.7 then implies the second assertion of Theorem 4.1, namely that n = 1.

Non-polynomial entire solutions of difference Super-Fermat functional equations
In this section we extend the results obtained in Sect. 4 for the case of entire solutions of hyper-order strictly less than one. The hyper-order of an entire function g is defined as where T (r , g) is the Nevanlinna characteristic function of g. For κ ∈ C\{0} we denote by P 1 κ the field of period κ meromorphic functions of hyper-order strictly less than one.
In the case of hyper-order ≥ 1, for an arbitrary integer n ≥ 2 there exists a transcendental entire function f (z) such that [ f ]n κ reduces to a constant. For example, consider f (z) = exp π(z)ω z/κ where π(z) is a κ-periodic entire function of order ρ(≥ 1) and ω = 1 is an nth root of unity. Then we have ρ 2 ( f ) = ρ and   2 and let f 1 , . . . , f m be non-constant entire functions such that 1 for all i ∈ {1, . . . , m}, and such that [ f 1

Example 5.3
It is well known that the Fermat functional equation . Corresponding to Proposition 5.2, the difference Fermat functional equation The Fermat functional equation where ℘ is the Weierstrass ℘-function satisfying (℘ ) 2 = 4℘ 3 −1. This parametrization of the Fermat curve is given by Gross [14], while the Weierstrass map ( 3 √ 4℘, ℘ ) parametrizes the elliptic curve y 2 = x 3 − 1. These uniformizations are equivalent, while functions appearing in both expressions behave differently from the value-distribution point of view. In fact, the four elliptic functions , 4℘ 3 and −(℘ ) 2 are all unramified outside {0, ∞, 1}. The former two functions attain the three values with the same multiplicity 3, while those multiplicities of the latter two function are 2, 6 and 3, both cases of which are extremal to Nevanlinna's inequality. Geometrically, we distinguish Gross' and Weiestrass' expressions in the above by the way of triangular tilings for the Euclidean plane.
Corresponding to Proposition 5.1, an elementary computation shows that the difference Fermat functional equation where ℘ is the Weierstrass ℘-function satisfying (℘ ) 2 = 4℘ 3 − 2 and z 0 and h are constants with ℘ (h) = 0.
Before we can prove Propositions 5.1 and 5.2, we need to introduce tools to handle entire functions. In particular, we will consider an extension of the notion of difference radical for entire functions, and define the corresponding Nevanlinna counting functions.
The order of a holomorphic curve g : C → P n is defined by where log + x = max{0, log x} for all x ≥ 0, and is the Cartan characteristic function of g with the reduced representation g = [g 0 : · · · : g n ].
Similarly, the hyper-order of g : C → P n is The following lemma [16,Lemma 8.3] is a useful tool in dealing with shifts in characteristic and Nevanlinna counting functions.
where r runs to infinity outside of a set of finite logarithmic measure.
We give a comparison estimate of T r , f (z + κ) in terms of T r ± |κ|, f (z) , as well as of N r , f (z + κ) in terms of N r ± |κ|, f (z) . In the proof of the following lemma we implement an alternate approach to the estimates discussed in [12, p. 47].
Remark One may notice that the second part can be simplified a little by observing the Ahlfors-Shimizu characteristic function instead of applying Cartan's identity to Nevanlinna's T (r , f ). See [29, Sections 3 and 4 of Chapter VI] or [31, Section 6], for example, for some related discussions on those characteristic functions as well as on the quantity 2π 0 n r , 1 We denote by D(z 0 , s) = {z ∈ C : |z − z 0 | ≤ s} the closed disc of radius s > 0 centred at z 0 ∈ C. We define, as in [6], the order ord ζ ( f ) of a meromorphic function f at ζ ∈ C as the unique μ ∈ Z such that With this notation ord ζ ( f ) > 0 if and only if f has a zero of order ord ζ ( f ) at ζ , and ord ζ ( f ) < 0 if and only if f has a pole of order −ord ζ ( f ) at ζ . We also adopt the notation as a difference analogue of the truncated counting function for the zeros of f . The corresponding integrated counting function is defined in the usual way as [q] Also, by defining The following lemma demonstrates how the truncation works with the counting function (5.5).
Proof Suppose first that n > q. Then by definition Each term in the minimum on the right hand side of equation (5.6) contains the sum To see this, we may write n−1 i=0 ord + w+(i+ j)κ ( f ) = n+ j−1 k= j ord + w+kκ ( f ), if necessary. Therefore, it follows by (5.6) that .
By integrating (5.7) it follows that since n > q. If n ≤ q, the inequality (5.8) holds trivially, so in fact we have (5.8) for all n ∈ N. The assertion now follows by Lemmas 5.4 and 5.5.
The following result is a truncated version of the second main theorem for differences proved in [16], and it can be viewed as a counterpart of Theorem 3.5 for entire functions. For the second main theorem of Cartan, we refer to [4] as well as [15]. Theorem 5.7 Let g 1 , . . . , g m be m ≥ 2 entire functions with no common zeros, linearly independent over P 1 κ , and let g m+1 = g 1 + · · · + g m . If the holomorphic curve g = [g 1 : · · · : g m ] satisfies ς(g) < 1, then where κ ∈ C\{0}, ε > 0, and r → ∞ outside of an exceptional set of finite logarithmic measure.
Suppose w is a zero of G. We assert that (5.10) To confirm this, we write . . .

Discussion
The Oesterlé-Masser abc conjecture for integers states that given ε > 0, there exists a number K (ε) having the following property: For any nonzero relatively prime integers a, b, c such that a + b = c, we have max{|a|, |b|, |c|} ≤ K (ε) rad(abc) 1+ε .
Another possible replacement of rad(abc) may be a sort of its difference analogue that we have studied in this note for functions. In other words, one can now ask the following: similarly as the Stothers-Mason theorem corresponds to the abc conjecture, what would be the analogue of Theorem 3.1 in number theory?
Let p 1 = 2, p 2 = 3, p 3 = 5, . . . , p i , . . . denote the collection of all prime numbers listed in increasing order, and let x ∈ N. Then we can write x in the unique form Recall that the sizes of rãd(x) and rad(x) are not in a fixed order such as rãd(2·3) < rad(2·3) and rãd(2 2 ) > rad(2 2 ). We have observed the same property for the corresponding radicals in the polynomial ring. Here let us observe one typical difference among their properties by seeing an interesting question on the ordinary radical in the integer ring that is known as the Erdös-Woods conjecture made by Woods in his Ph. D. Thesis [44]: There exists an absolute constant k > 2 such that for every positive integers x and y, the conditions rad(x + i) = rad(y + i), i = 0, 1, 2, . . . , k − 1 imply x = y. Erdös gave a counterexample x = 75 = 3 · 5 2 and y = 1215 = 3 5 · 5 showing the necessity of the assumption k > 2, since 76 = 2 2 · 19 and 1216 = 2 6 · 19. By the idea used above, one also finds the pair x = 2 m+1 − 2 and y = 2 m+1 (2 m+1 − 2) for each m ∈ N, since rad 2 m+1 (2 m+1 − 2) + 1 = rad(2 m+1 − 1). On the other hand, any such examples seem not possible at all for our difference radical. For example, we have rãd(3 · 5 2 ) = 5 2 , rãd(3 5 · 5) = 3 4 · 5, rãd(2 2 · 19) = 2 2 · 19 and rãd(2 6 · 19) = 2 6 · 19, and also rãd(2 m+1 − 2) = rãd 2 m+1 (2 m+1 − 2) , rãd(2 m+1 − 1) = rãd (2 m+1 − 1) 2 for each m ∈ N, respectively. Of course, it would be imprudent of us to claim that rãd has a higher accuracy than rad does only with this kind of simple comparison. It is reasonable for us to say instead that each of both radicals demonstrates their own ability in leveling the multiple factors of target in the ring under observation, that is, of polynomials or of integers, respectively. Hence we believe that the new radical rãd gives at least another indicator of the factorization of integers and thus a difference analogue of the abc conjecture, which might have been stated as follows: For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rãd(abc) 1+ε .
No three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. Of course, we do not accept 3 · 4 + 5 · 6 = 6 · 7 as an example for n = 2 since this reduces to 2 + 5 = 7 (and moreover 4 and 6 are not primes). It is trivial that a 1 +b 1 = c 1 has a solution (a, b, c) consisting of 1 or prime numbers, when and only when it is the triple of 2 and an odd prime pair except for trivial 1+1 = 2 and 1+2 = 3 then. Thus the existence of infinitely many of such solutions would imply the existence of infinitely many twin primes, which is still an open question. For n = 2, it seems difficult to determine such prime solutions ( p i , p j , p k ), that is, those satisfying p i p i+1 + p j p j+1 = p k p k+1 . One finds at least the triple (1, 5, 2 2 ) including a single prime with 1 and satisfying 1 2 + 5 2 = (2 2 ) 2 .
In order to remind ourselves of the way of our shift, let us note that the triple (1, 11, 3 2 ) is not such a solution for n = 2, since (3 2 ) 2 is not (3 · 4) 2 but (3 · 5) 2 . Our shift step κ is however allowed to be strictly greater than one or even to be a negative integer, so the triple represented as (1, 13, 2 2 · 3 2 ) can be our solution for n = 2 of shift −1 with p 0 = 1, that is, 1 + 13 × 11 = 2 2 · 3 2 × 1 2 · 2 2 . In each case, there is however an artificial treatment of the number p 0 = 1, so that we are eager for a spontaneous solution at least for n = 2 with some shift step κ.
In Section 3 we have implemented an approach by Shapiro and Sparer [32] in assuming that the polynomials in Theorem 3.4 are pairwise relatively prime. Extending this result to the case where polynomials in any vanishing sub-sum are relatively prime using the method of de Bondt [8] is left as an open question. In [8] one finds also a generalization of the Davenport-Stothers inequality. We also leave the problem of extending our results to polynomials or holomorphic functions of more than two complex variables as open questions. Some related discussions and good lists of references are found in Hu and Yang's monograph [23,Chapter 5] as well as [22,Section 3.2].
Last but not least, we mention Vaserstein's paper [41] where he proved a quantum abc theorem for polynomials over an algebraically closed number field. Theorem 3.1 corresponds to the "h-version" of [41, Theorem 1.1] with different notation and proof. Again, we leave finding "(h, q)-versions" with q = 1 for the remaining results in our paper as a future project.