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EnglishFor positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure of a finite prime field Fp, as well as L-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions φ1,…,φm,ϱ1,…,ϱn ∈ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many α in the algebraic closure of F_p, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (ϱ1(α),⋅),…,(ϱn(α),⋅) are L-linearly dependent on the reduction of E modulo p. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety G^k_m×E^n with the algebraic subgroups of codimension at least 2. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases.
Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes / Barroero, Fabrizio; Capuano, Laura; Mérai, Laszlo; Ostafe, Alina; Sha, Min. - ELETTRONICO. - (2020).
| Titolo: | Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes | |
| Autori: | ||
| Data di pubblicazione: | 2020 | |
| Abstract: | For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure of a finite prime field Fp, as well as L-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions φ1,…,φm,ϱ1,…,ϱn ∈ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many α in the algebraic closure of F_p, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (ϱ1(α),⋅),…,(ϱn(α),⋅) are L-linearly dependent on the reduction of E modulo p. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety G^k_m×E^n with the algebraic subgroups of codimension at least 2. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases. | |
| Appare nelle tipologie: | 5.12 Altro |
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http://hdl.handle.net/11583/2849211
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