key: cord-0637847-jk1rjshy
authors: Ostafe, Alina
title: On a Problem of Lang for Matrix Polynomials
date: 2021-05-17
journal: nan
DOI: nan
sha: c77c6741778b448c0811a01785343164fc11ada3
doc_id: 637847
cord_uid: jk1rjshy

In this paper, we consider a problem of Lang about finiteness of torsion points on plane rational curves, and prove some results towards a matrix analogue of this problem, including a full analogue for $2times 2$ matrices defined over $mathbb{C}$.

1.1. Motivation. Pivotal work of Lang made it clear that the existence of multiplicative relations between coordinates of points on algebraic curves in G n m = (C {0}) n is a very rare event, which may occur only if the curve is "special". In particular, the celebrated result conjectured by Lang [9, 13] in the 1960s and proved by Ihara, Serre and Tate asserts the finiteness of so-called torsion points on curves, that is, points with all coordinates roots of unity. For the case of plane curves, Beukers and Smyth [2, Section 4.1] give a uniform bound for the number of such points, and Corvaja and Zannier [7] give an upper bound for the maximal order of torsion points on the curve. More precisely, one has the following result [2, Section 4.1]:

Theorem A. An algebraic curve F (y 1 , y 2 ) = 0, where F ∈ C[y 1 , y 2 ], contains at most 11(deg F ) 2 torsion points unless F has a factor of the form y i 1 − ρy j 2 or y i 1 y j 2 − ρ for some nonnegative integers i, j not both zero and some root of unity ρ.

Theorem A in the case of plane rational curves can be reformulated as follows: given multiplicatively independent rational functions f, g ∈ C(x) (see below for the precise definition), there are at most

elements α ∈ C such that both f (α) and g(α) are roots of unity, see also the proof of [11, Lemma 2.2] . This has been extended to a finiteness result of elements α ∈ C such that |f (α)| = |g(α)| = 1, first by Corvaja, Masser and Zannier [5] for f (x) = x and g ∈ C[x], and later by Pakovich and Shparlinski [12] for the general case, improving also the bound above for genus zero curves. More precisely, we have the following result [12, Theorem 2.2]:

Theorem B. Let f, g ∈ C(x). Then one has #{α ∈ C : |f (α)| = |g(α)| = 1} ≤ (deg f + deg g) 2 ,

unless f = f 1 • h and g = g 1 • h 1 for some quotients of Blaschke products f 1 and f 2 and some rational function h.

As remarked in [12] (see the comment after Theorem 2.2 in [12] ), if f and g are polynomials, then the conclusion of Theorem B holds, unless the polynomials f and g are multiplicatively dependent.

In this note, we aim at obtaining an analogue of Theorem A (for plane rational curves) for matrix polynomials.

Notation and conventions: We now set the following notation, which remains fixed for the remainder of this paper:

• For r ≥ 1, M r (C) is the set of all r × r matrices with entries in C, GL r (C) the set of invertible matrices, and SL r (C) the set of matrices of determinant one. • I ∈ M r (C) is the identity matrix.

• We use 0 for both the zero scalar and the zero matrix, which shall be clear from the context. • By a scalar matrix we mean a scalar multiple of the identity I, that is, λI for some λ ∈ C. • x, y 1 , y 2 are 'scalar' variables, that is, we apply them at elements λ ∈ C.

We reserve Z, Z 1 , Z 2 for 'matrix' variables, that is, we apply them at matrices A ∈ M r (C). We also write xI for the multiplication of the variable x with the identity matrix I. • f, g ∈ M r (C)[Z] are matrix polynomials with coefficients in M r (C), that is, polynomials of the form

is called a torsion point in GL r (C) 2 if both matrices A and B are torsion.

We say that two matrices A, B ∈ M r (C) are conjugate if there exists an invertible matrix V ∈ M r (C) such that

Clearly, two conjugate matrices have the same set of eigenvalues with the same multiplicities.

We say that two algebraic functions h 1 , h 2 ∈ C(x) are multiplicatively dependent if there is a non-zero pair of integers (k 1 , k 2 ) ∈ Z 2 such that

Otherwise they are called multiplicatively independent.

As a direct consequence of Theorem B, one already has an immediate result for matrix polynomials f, g ∈ M r (C) [Z] such that all the eigenvalues of f (λI) and g(λI), λ ∈ C, are of absolute value one. More precisely, one has:

be such that det(f (xI)) and det(g(xI)) are multiplicatively independent in C(x). Then there are at most

elements λ ∈ C such that f (λI) and g(λI) satisfy | det(f (λI))| = | det(g(λI))| = 1.

In particular, there are at most finitely many elements λ ∈ C such that all eigenvalues of f (λI) and g(λI) are of absolute value one.

Remark 1.2. The condition that det(f (xI)) and det(g(xI)) are multiplicatively independent in C(x) in Corollary 1.1 can be reformulated as follows: there is no non-zero pair of integers (k 1 , k 2 ) ∈ Z 2 such that

Indeed, det(f (xI)) and det(g(xI)) are multiplicatively independent in C(x) if and only if there is no non-zero pair of integers (k 1 , k 2 ) ∈ Z 2 such that det(f (xI)) k 1 det(g(xI)) k 2 = det f (xI) k 1 g(xI) k 2 = 1, which implies the above condition.

We also note that if f, g ∈ C[Z], then for any matrix A ∈ M r (C), by the spectral theorem on eigenvalues, the eigenvalues of f (A) are f (λ i ), i = 1, . . . , r, where λ 1 , . . . , λ r are the eigenvalues of A, and similarly for g. Thus, if f (A) n = I for some n, then all f (λ i ), i = 1, . . . , r, are roots of unity, and similarly for g. We reduce thus the problem to the classical Lang problem, that is, Theorem A. Similarly, if all eigenvalues of f (A) and g(A) are of absolute value one, then we reduce the problem to Theorem B.

If f, g ∈ M r (C)[Z] with coefficients C i = c i I, i = 1, . . . , deg f , and similarly for g, then we are in the case above, that is, f ∈ C[Z] is given by

and similarly for g, and thus the discussion above applies, again.

Theorem A is also intimately related to the question of giving uniform bounds for the degree of gcd(f n − 1, g m − 1), n, m ≥ 1, for some polynomials f, g ∈ C[x], which was initially considered by Ailon and Rudnick [1] and later in [11] and further extended in several ways by other authors. It is worth mentioning that matrices have already been considered in this context in [1] , that is, the authors give results for gcd(A n − I), n ≥ 1, for a matrix A defined over Z, cyclotomic extensions or C[T ] (here, by the greatest common divisor of a matrix we mean the greatest common divisor of all entries of the matrix). Moreover, in [6] , Corvaja, Rudnick and Zannier study the growth of the order of matrices in reduction modulo integers N ≥ 1 as N goes to infinity.

We note that the finiteness result in Theorem A has been extended to higher order multiplicative relations of points on curves in G n m defined over Q by Bombieri, Masser and Zannier [3] , and then further generalised in [4, 10] .

We conclude this section with a rather vague question towards obtaining a full matrix analogue of Theorem A for torsion points on plane curves.

Under what conditions on F are there, up to conjugacy, finitely many torsion points (A 1 , A 2 ) ∈ GL r (C) 2 such that F (A 1 , A 2 ) = 0?

In this paper, we give an answer for the 2×2 matrix analogue of Theorem A in the case of plane rational curves.

1.2. Main results. Informally, given matrix polynomials f, g ∈ M r (C)[Z], we would like to have a finiteness result for the set of matrices A ∈ M r (C), such that f (A) and g(A) are "roots" of the identity matrix. In this paper, we are able to prove this in any dimension r for matrices A ∈ M r (C) that commute with the coefficients of both f and g, as well as for arbitrary matrices A ∈ M 2 (C) in dimension two.

It is clear that, in the case of matrices, one cannot expect a finiteness result as in Theorem A. Indeed, let f have the coefficients c i I, c i ∈ C, i = 0, . . . , deg f , and let A ∈ M r (C) be such that f (A) n = I for some n. Then any matrix conjugate to A is also a solution to f (Z) n = I, and similarly for g. Thus, one can only expect a finiteness result up to conjugacy.

Our first result gives an answer towards Lang's problem for matrices which commute with the coefficients of the polynomials f and g. More precisely, we have:

be such that any eigenvalue of f (xI) and any eigenvalue of g(xI) are multiplicatively independent functions in C(x). Then, up to conjugacy, there are at most

The proof reduces to considering scalar specialisations, see Lemma 2.2 (in Section 2.2), and thus relies on Theorem A above.

As an example, one can consider all coefficients of f and g to be matrices in C[B] for some fixed B ∈ M r (C). Then Theorem 1.4 gives finiteness, up to conjugacy, of the set of matrices A ∈ M r (C) which commute with B, such that (f (A), g(A)) is a torsion point.

The main result of the paper is a full analogue of Lang's result (in the case of plane rational curves) for 2 × 2 complex matrices. To state our result, we introduce the following notation and definition: for f ∈ M 2 (C)[Z], we define the set

where µ i (x), i = 1, . . . , r, are the eigenvalues of f (xI) in C(x).

Definition 1.5. We say that two polynomials f, g ∈ M 2 (C) [Z] are spectrally multiplicatively independent if for any pair (α, β) ∈ S f × S g , where S f and S g are defined by (1.1), α and β are multiplicatively independent.

Remark 1.6. We note that any eigenvalue µ i (x) of f (xI) being multiplicatively independent with any eigenvalue η j (x) of g(xI) would not necessarily imply that det(f (xI)) is multiplicatively independent with all η j (x), j = 1, . . . , r, or that det(f (xI)) is multiplicatively independent with det(g(xI)). We need the latter conditions to apply Corollary 1.1 or Lemma 2.5 in the proof of our main result, Theorem 1.7 below, and thus the need to include det(f (xI)) =

We have the following:

be spectrally multiplicatively independent and such that f (xI) and g(xI) are nonsingular. Then, up to conjugacy, there are at most

The proof of this result is based on [8, Theorem 1] (see Section 2.3), coupled with Corollary 1.1 above, Lemma 2.2 and Lemma 2.5 (see Section 2.2). Remark 1.8. We note that to ensure that f (xI) and g(xI) are nonsingular, it is enough to assume, for example, that the leading matrix coefficients of f and g are non-singular matrices.

We expect that the spectral multiplicative independence condition in Theorem 1.7 holds for the overwhelming majority of pairs of matrix polynomials f and g.

We also remark that, if all the eigenvalues of f (xI) and g(xI) are multiplicatively independent functions in C(x), then the spectral multiplicative independence condition is satisfied. For example, let

be such that a 1 , a 3 , b 1 , b 3 ∈ C * are multiplicatively independent. We note that the eigenvalues of f (xI) are x d + a i , i = 1, 3, and similarly the eigenvalues of g(xI) are x e + b i , i = 1, 3. Since a 1 , a 3 , b 1 , b 3 ∈ C * are multiplicatively independent, all conditions of Theorem 1.7 are satisfied.

We obtain the following consequence of Theorem 1.7.

be such that C is non-singular. Then, up to conjugacy, there are at most 2 27 torsion points

Remark 1.10. We note that indeed Corollary 1.9 is not necessarily true if det(C) = 0. For example, let

For any primitive p-th root of unity λ, the point

is torsion of order p. Indeed, this follows immediately since

Therefore, we have infinitely many such matrices (which are not similar) when λ runs over all p-th roots of unity, p prime.

In Theorem 1.4 and Theorem 1.7, we look at matrices A ∈ M r (C) such that all the eigenvalues of f (A) and g(A) are roots of unity. We would also like to have a more general result for the case when all the eigenvalues of f (A) and g(A) are of absolute value one. This, then, would be an analogue of Theorem B and would extend Corollary 1.1 to non-scalar matrices. We thus formulate the following problem:

. Prove that, under certain conditions on f and g, there are, up to conjugacy, finitely many matrices A ∈ M r (C) such that all the eigenvalues of f (A) and g(A) are of absolute value one.

and the following two resultants

We note that both R f,g and T f,g are non-zero polynomials. Indeed, assume that R f,g = 0. Then, by the definition of the resultant, the polynomials P f (x, y 1 ) and P g (x, y 2 ), as polynomials in x, share a common root t ∈ C(y 1 ) ∩ C(y 2 ) = C. Thus we obtain that det(y 1 I − f (tI)) = det(y 2 I − g(tI)) = 0, which is a contradiction, since both polynomials have as leading monomials y r 1 and y r 2 , respectively. Similarly, T f,g is a non-zero polynomial.

We know that deg x P f ≤ r deg f and deg x P g ≤ r deg g, and R f,g is a polynomial of degree deg x P f in y 2 and of degree deg x P g in y 1 . Similarly, deg det(g(xI)) ≤ r deg g, and T f,g is a polynomial of degree deg x P f in y 2 and of degree deg det(g(xI)) in y 1 . We thus obtain that

If any eigenvalue of f (xI) and any eigenvalue of g(xI) are multiplicatively independent functions in C(x), then R f,g (y 1 , y 2 ) defined by (2.2) does not have a factor of the form y i 1 y j 2 − ρ or y i 1 − ρy j 2 for some nonnegative integers i, j not both zero and some root of unity ρ.

(ii) If any eigenvalue of f (xI) and det(g(xI)) are multiplicatively independent functions in C(x), then T f,g (y 1 , y 2 ) defined by (2.2) does not have a factor of the form y i 1 y j 2 − ρ or y i 1 − ρy j 2 for some nonnegative integers i, j not both zero and some root of unity ρ.

Proof. The proofs for (i) and (ii) follow the same discussion, so we only provide the proof for (i).

Let µ i (x), i = 1, . . . , r, be the eigenvalues of f (xI) in C(x), that is, the roots of the polynomial P f (y 1 , x) defined by (2.1) as a polynomial in y 1 . Similarly, let η j (x), j = 1, . . . , r, be the eigenvalues of g(xI) in C(x).

Assume that R f,g (y 1 , y 2 ) has a factor of one of the forbidden forms, say y i 1 y j 2 − ρ for some nonnegative integers i, j not both zero and some root of unity ρ. We note that any point on the curve R f,g (y 1 , y 2 ) = 0 is of the form (µ k (t), η ℓ (t)) for some 1 ≤ k, ℓ ≤ r and some t ∈ C. Indeed, let (t 1 , t 2 ) ∈ C 2 be such that R f,g (t 1 , t 2 ) = 0. Then, by definition of the resultant R f,g , the two polynomials

have a common root x = t ∈ C. This implies that t 1 = µ k (t) and t 2 = η ℓ (t) for some k, ℓ. Since y i 1 y j 2 − ρ is a factor of R f,g , there are infinitely many (t 1 , t 2 ) ∈ C 2 which are roots of this factor, and thus we deduce that there are infinitely many t ∈ C such that µ k (t) i η ℓ (t) j = ρ for some 1 ≤ k, ℓ ≤ r. Since µ k and η ℓ are algebraic functions, we conclude that µ k (x) i η ℓ (x) j = ρ, which contradicts our hypothesis.

The case when R f,g (y 1 , y 2 ) has a factor of the form y i 1 −ρy j 2 is treated entirely similar. A similar discussion applies for (ii), replacing only det(t 2 I − g(xI)) above with the polynomial t 2 − det(g(xI)).

Two of the main tools for the proof of Theorems 1.4 and 1.7 are the following results which apply, again, to scalar matrices λI, however for which the matrices f (λI) and g(λI) satisfy different conditions than in Corollary 1.1. More precisely, we have:

be such that any eigenvalue of f (xI) and any eigenvalue of g(xI) are multiplicatively independent functions in C(x). Then there are at most

elements λ ∈ C such that f (λI) n − I and g(λI) m − I are singular matrices for some n, m ≥ 1.

Proof. We use a similar approach as for the proof of [1, Theorem 3], reducing the problem to an application of Theorem A.

Let λ ∈ C be such that f (λI) n − I and g(λI) m − I are singular matrices for some n, m ≥ 1. This implies that J n f (λI) and J m g(λI) , which are triangular matrices, have at least one element 1 on the main diagonal, where J f (λI) and J g(λI) are Jordan normal forms of f (λI) and g(λI), respectively.

Let u λ,i , v λ,j ∈ C, i, j = 1, . . . , r, be the eigenvalues of f (λI), g(λI), respectively, that is, u λ,i are the (not necessarily distinct) roots of the polynomial P f (λ, y 1 ) and v λ,j are the (not necessarily distinct) roots of the polynomial P g (λ, y 2 ), where P f (x, y 1 ) and P g (x, y 2 ) are defined by (2.1). Consequently, there exist i, j ∈ {1, . . . , r} such that u n λ,i = 1 and v m λ,j = 1, that is, both u λ,i and v λ,j are roots of unity.

Notice that, since P f (λ, u λ,i ) = P g (λ, v λ,j ) = 0, one also has R f,g (u λ,i , v λ,j ) = 0 for all i, j, where R f,g is defined by (2.2). Moreover, from the above discussion, there exist i, j such that (u λ,i , v λ,j ) is a torsion point on the curve R f,g (y 1 , y 2 ) = 0. Since, by our hypothesis and Lemma 2.1 (i), R f,g does not have any of the special factors mentioned in the statement of Theorem A, it follows from Theorem A and (2.3) that there are at most 11(deg R f,g ) 2 ≤ 11r 2 (deg f + deg g) 2 torsion points (ζ 1 , ζ 2 ) on the curve R f,g (y 1 , y 2 ) = 0. Each such point (ζ 1 , ζ 2 ) = (u λ,i , v λ,j ) for some i, j corresponds to at most r min(deg f, deg g) values of λ. Indeed, since R f,g (ζ 1 , ζ 2 ) = 0, λ is a common root of the polynomials P f (x, ζ 1 ), P g (x, ζ 2 ). We note that both polynomials P f (x, ζ 1 ), P g (x, ζ 2 ) are non-zero, since, otherwise, ζ 1 or ζ 2 would be an eigenvalue of f (xI) or g(xI), respectively. However, since ζ 1 or ζ 2 are roots of unity, this contradicts the multiplicative independence assumption on the eigenvalues of f (xI) and g(xI).

Taking the contribution from each i, j ≤ r, we conclude that there at most

possibilities for such λ ∈ C, which concludes the proof. 

elements λ ∈ C such that f (λI) and g(λI) have each at least one eigenvalue that is a root of unity.

Remark 2.4. When r = 1, the conditions in Corollary 1.1 and Lemma 2.2 are equivalent to the polynomials f and g being multiplicatively independent, and, in this case, we recover Theorem A.

be such that any eigenvalue of f (xI) and det(g(xI)) are multiplicatively independent functions in C(x). Then there are at most

for some n, m ≥ 1.

Thus, in Lemma 2.5, we look at λ ∈ C such that f (λI) has an eigenvalue a root of unity and det(g(λI)) is a root of unity.

Proof. The proof follows exactly the same lines as the proof of Lemma 2.2, but, instead of the polynomial R f,g , we consider T f,g defined by (2.2).

Indeed, let λ ∈ C be such that f (λI) n − I is singular for some n ≥ 1 and det(g(λI)) is a root of unity. As observed above, this means that an eigenvalue of f (λI) is a root of unity, which we denote, as in the previous proof, by u λ,i for some i = 1, . . . , r. Since T f,g (u λ,i , det(g(λI))) = 0, we are, again, in the situation of looking at torsion points on the algebraic curve T f,g (y 1 , y 2 )=0 and apply Theorem A. Using (2.3) and applying Lemma 2.1 (ii) and Theorem A, we obtain at most

torsion points (ζ 1 , ζ 2 ) on the curve T f,g (y 1 , y 2 ) = 0. Each such torsion point (ζ 1 , ζ 2 ) = (u λ,i , det(g(λI))) for some i = 1, . . . , r, corresponds, again, to at most r min(deg f, deg g) values of λ. Taking the contribution from each i ≤ r, we conclude that there at most

possibilities for such λ ∈ C, which concludes the proof.

In this section, we consider only 2 × 2 matrices. For matrices A, B ∈ GL 2 (C), we define the set

In [8, Theorem 1], Evertse and Tijderman give a classification of pairs of matrices (A, B) such that the set S A,B is infinite. This is our main tool in the proof of Theorem 1.7. For completeness, we present their result in this section, and, for this, we say that two pairs of matrices (A, B) and (A 1 , B 1 ) are similar, if there exists a matrix V ∈ GL 2 (C) such that

Moreover, we say that

. We define now four pairs of matrices (A 1 , B 1 ) for which S A 1 ,B 1 is infinite, see [8] for more details.

for some integers ℓ, s not both zero, and some non-zero θ ∈ C.

(II) A ℓ 1 = θ 0 0 κ and B s 1 = 0 ζ ζ 0 for some integers ℓ, s with ℓs = 0 and some non-zero θ, κ, ζ ∈ C such that θκ = ζ 2 .

for some integers ℓ, s with ℓs = 0 and some non-zero θ, κ, ζ ∈ C such that κζ = θ 2 .

(1 + √ ζµ)ρ , for some α, ρ, ζ, µ ∈ C such that µ = 0, α and ρ are not roots of unity, and (α n − ρ m ) 2 = µnmα n ρ m for infinitely many (n, m) ∈ Z 2 .

Remark 2.6. We note that for pairs (A 1 , B 1 ) of type IV above, both A 1 and B 1 have a double eigenvalue, namely α and ρ, respectively.

We Using the commutativity assumption on A, simple computations show that there exist polynomials Q n,A , Q m,A ∈ M r (C) depending on n, m and A, such that

Therefore, using (3.1), we obtain that det(xI − A) | gcd (det(f (xI) n − I), det(g(xI) m − I)) .

We note that both polynomials det(f (xI) n − I) and det(g(xI) m − I) are non-zero. Indeed, assume, for example, that det(f (xI) n − I) = 0. Then writing

where ζ ∈ C is an n-th root of unity, we conclude that det(f (xI)−ζ i I) = 0 for some i = 1, . . . , n. Thus ζ i is an eigenvalue of f (xI), and similarly for g. This contradicts our multiplicative independence assumption on the eigenvalues of f (xI) and g(xI).

Thus, every eigenvalue of A is a root of the greatest common divisor above. In other words, for any eigenvalue λ ∈ C of A, the matrices f (λI) n − I and g(λI) m − I are singular. The conclusion now follows from Lemma 2.2, that is, there are at most L = 22r 5 (deg f + deg g) deg f · deg g possibilities for each of the eigenvalues of A.

We partition now the set {1, . . . , r} into k ordered parts, 1 ≤ k ≤ r, where each such part corresponds to a Jordan block of A, and thus to one eigenvalue λ. The number of such partitions is r−1 k−1 , and each set in a partition corresponds to at most L values of λ ∈ C. Summing over all k we obtain at most

possible Jordan normal forms, which concludes the proof.

Proof of Theorem 1.7. We start by remarking that the spectral multiplicative independence assumption ensures that the conditions in Corollary 1.1 and Lemmas 2.2 and 2.5 are satisfied, and thus we can apply these results, see the end of the proof. Let A ∈ M 2 (C) be such that Let λ be an eigenvalue of A and v the corresponding eigenvector, that is, one has

We note that, if f (λI) is singular, then this implies that det(f (λI)) = 0, that is, λ is a zero of a non-zero polynomial (by our hypothesis) of degree at most 2 deg f . Thus, there are at most 2 deg f such elements λ, which we exclude from the discussion below. The same discussion applies for g(λI), thus, from now on, we assume that both f (λI) and g(λI) are non-singular.

The idea of the proof is to show that the sets S f (A),f (λI) and S g(A),g(λI) defined by (2.4) are infinite. Then, applying Theorem 2.7, we obtain that (f (A), f (λI)) and (g(A), g(λI)) are related to pairs of matrices of type I, II or III as defined in Section 2.3 (we will see that the type IV cannot occur). This will allow us to conclude that one of f (λI) and g(λI) has an eigenvalue which is a root of unity, while the other matrix will have same property or the product of its eigenvalues is a root of unity. Applying, then, Corollary 1.1, Lemma 2.2 or Lemma 2.5, we will conclude that there are finitely many such λ ∈ C. Since this discussion applies for any eigenvalue λ of A, we conclude the proof.

First, we remark that, using (3.3), for any integer i ≥ 1, one has

which implies that

or, equivalently,

Since v ∈ C 2 is a non-zero vector, we conclude that the matrix f (A) − f (λI) is singular, and similarly for g. Moreover, using our hypothesis (3.2), we obtain, for any integer k ≥ 1,

Thus, for any integer k ≥ 1, the matrices

are singular, which implies that the sets S f (A),f (λI) and S g(A),g(λI) defined by (2.4) are infinite. Therefore, Theorem 2.7 tells us that (f (A), f (λI)) and (g(A), g(λI)) are related to pairs of type I, II, III or IV as defined in Section 2.3. We only consider the pair (f (A), f (λI)), a similar argument also applies to (g(A), g(λI)).

Let (f (A), f (λI)) be related to (A 1 , B 1 ) of type I, II, III or IV, which means that (f (A), f (λI)) is similar to one of (A 1 , B 1 ), (B 1 , A 1 ), (A T 1 , B T 1 ) or (B T 1 , A T 1 ).

(I) We assume first that (f (A), f (λI)) is similar to (A 1 , B 1 ), where the pair (A 1 , B 1 ) is such that

for some integers ℓ, s not both zero, and some non-zero θ ∈ C. Using (3.2), since f (A) ℓn = I is similar to A ℓn 1 , we obtain that θ is an n-th root of unity (we note that, if ℓ = 0, then θ = 1).

Since f (λI) is similar to B 1 , and θ is an eigenvalue of B s 1 , we conclude that f (λI) s has an eigenvalue θ which is a root of unity, and, thus, f (λI) has also an eigenvalue which is a root of unity.

We note that a similar discussion applies for the case when (f (A), f (λI)) is similar to one of (B 1 ,

, which concludes this case. (II) (i) We assume first that (f (A), f (λI)) is similar to (A 1 , B 1 ), where the pair (A 1 , B 1 ) is such that

for some integers ℓ, s with ℓs = 0 and some non-zero θ, κ, ζ ∈ C such that

The same discussion as for Case (I) concludes that both θ and κ are n-th roots of unity, and thus, by (3.4) , ζ is also a root of unity. Since ζ and −ζ are the eigenvalues of B s 1 , and f (λI) is similar to B 1 , we conclude, again, that both eigenvalues of f (λI) are roots of unity, and thus det(f (λI)) is a root of unity.

(ii) We assume now that (f (A), f (λI)) is similar to (B 1 , A 1 ). As in the previous discussions, since f (A) is similar to B 1 and ζ is an eigenvalue of B s 1 , we conclude that ζ is a root of unity. Using the relation (3.4), we conclude that θκ is a root of unity, and, thus, so is det(A ℓ 1 ). Therefore, det(A 1 ) is a root of unity.

Moreover, since f (λI) is similar to A 1 , we conclude that det(f (λI)) = det(A 1 ), and thus det(f (λI)) is also a root of unity.

A similar discussion applies for the case when (f (A), f (λI)) is similar to one of (A T 1 , B T 1 ), (B T 1 , A T 1 ), which concludes this case. (III) We assume first that (f (A), f (λI)) is similar to (A 1 , B 1 ), where the pair (A 1 , B 1 ) is such that

for some integers ℓ, s with ℓs = 0 and some non-zero θ, κ, ζ ∈ C such that κζ = θ 2 .

The same considerations as for Cases (I) and (II) (i) apply, and thus we obtain that both θ and κ are roots of unity, which in turn implies that ζ is also a root of unity. Noting now that the eigenvalues of B s 1 are ζ and −θ, we conclude that the eigenvalues of B 1 , and thus of f (λI), are roots of unity, and thus det(f (λI)) is a root of unity. A similar discussion applies for the case when (f (A), f (λI)) is similar to one of (B 1 , A 1 ), (A T 1 , B T 1 ), (B T 1 , A T 1 ), which concludes this case. (IV) If f (A) is similar to A 1 or A T 1 , then α is a root of unity, which contradicts the assumption in (IV) in Section 2.3. Thus, we can only have that (f (A), f (λ)) is similar to (B 1 , A 1 ) or (B T 1 , A T 1 ). However, this, again, implies that ρ is an eigenvalue of f (A), and thus a root of unity, which is not possible.

Similarly as above, one concludes that either g(λI) has one eigenvalue which is a root of unity (as in case (I)) or the product of its eigenvalues, and thus det(g(λI)), is a root of unity (as in cases (II) and (III)).

To conclude the finiteness of the set of λ ∈ C as above, we consider all possible combinations for f (λI) and g(λI) in the cases (I), (II) and (III). For each combination, we obtain the following bounds for the cardinality of the set of such λ ∈ C:

• If both f (λI) and g(λI) have each one eigenvalue a root of unity (occurring in case (I)), then, by Lemma 2.2, we obtain at most

possibilities for each of the eigenvalues of such A. • If | det(f (λI))| = | det(g(λI))| = 1 (occurring when both (f (A), f (λI)) and (g(A), g(λI)) fall in any of the cases (II) and (III)), we apply Corollary 1.1 to obtain at most 2 2 (deg f + deg g) 2 possibilities for each of the eigenvalues of such A.

• When (f (A), f (λI)) falls in case (I) and (g(A), g(λI)) falls in one of the cases (II) or (III), or the other way around, we apply Lemma 2.5 to obtain 22 · 2 4 (deg f + deg g) deg f · deg g possibilities for each of the eigenvalues of such A.

Thus, this case contributes in total with 22 · 2 5 (deg f + deg g) deg f · deg g possibilities for each of the eigenvalues of A in this case. Taking also into account the contribution of at most 2 2 deg f deg g elements λ ∈ C for which det(f (λI)) = 0 or det(g(λI)) = 0, which we excluded at the beginning of the proof, and putting everything together, we obtain at most 2 22 · 2 5 (deg f + deg g) deg f · deg g + 2 2 (deg f + deg g) 2 + 2 2 deg f deg g We conclude now the proof by observing, as in the proof of Theorem 1.4, that there are at most J(J + 1) ≤ 2J 2 ≤ 2 25 (deg f + deg g) 2 (deg f · deg g) 2 possible Jordan forms.

3.3. Proof of Corollary 1.9. The result follows directly from Theorem 1.7 applied to the polynomials f (Z) = Z and g(Z) = Z − C. The determinants of f (xI) and g(xI) are given by det(f (xI)) = x 2 and det(g(xI)) = x 2 − tr(C)x + det(C),

where tr(C) is the trace of the matrix C. Since det(C) = 0, the two determinants are multiplicatively independent.

The eigenvalue of f (xI) is x with multiplicity two, and a simple computation shows that the eigenvalues of g(xI) are given by 2x − tr(C) ± tr(C) 2 − 4 det(C) /2.

We notice that these latter eigenvalues are multiplicatively independent with x, and thus with det(f (xI)) as well, since, again, det(C) = 0. The bound now follows from Theorem 1.7, which concludes the proof.

Torsion points on curves and common divisors of a k − 1 and b k − 1

Cyclotomic points on curves

Intersecting a curve with algebraic subgroups of multiplicative groups

On unlikely intersections of complex varieties with tori

Sharpening 'Manin-Mumford' for certain algebraic groups of dimension 2

A lower bound for periods of matrices

Singular differences of powers of 2 × 2-matrices

Fundamentals of Diophantine Geometry

Courbes algébriques etéquations multiplicatives

On some extensions of the Ailon-Rudnick theorem

Level curves of rational functions and unimodular points on rational curves

Lecture notes on Diophantine analysis

The author is very grateful to Daniel Raoul Perez, Zeev Rudnick, Igor Shparlinski and Umberto Zannier for many discussions and useful comments on preliminary versions of the paper, which improved it significantly. The author thanks Umberto Zannier for suggesting the formulation of Question 1.3 and considering Corollary 1.9. The author is also grateful to the referee for their useful comments.The author was partially supported by the Australian Research Council Grant DP200100355. The author also gratefully acknowledges the hospitality and generosity of the Max Planck Institute of Mathematics (which fully supported the author for 11 months in 2020, during a very challenging Covid-19 time), where this project was initiated.