Additive and multiplicative relations with algebraic conjugates

• Artūras Dubickas ^[1] ; Paulius Virbalas ^[1]
  1. [1] Vilnius University

     Vilnius University

     Lituania

     
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 69, Nº. 1, 2026, págs. 143-154
• Idioma: inglés
• DOI: 10.33044/revuma.5106
• Enlaces
  • Texto completo
• Resumen

  • We prove that every nontrivial additive relation between algebraic conjugates of degree d over Q has a corresponding multiplicative relation. The proof is constructive. The reverse statement is not true. These findings supplement the research of Smyth, Dixon, Girstmair and others. In addition, following a result of Kitaoka we show that all additive relations between four distinct algebraic conjugates of degree 4 over Q can be described as Z-linear combinations of several basic nontrivial relations. On the other hand, we prove that an analogous result no longer holds for algebraic conjugates of degree 6 over Q .

• Referencias bibliográficas
  • E. Artin, Galois theory, second ed., Dover, Mineola, NY, 1998. MR Zbl
  • G. Baron, M. Drmota, and M. Skałba, Polynomial relations between polynomial roots, J. Algebra 177 no. 3 (1995), 827–846. DOI MR Zbl
  • J. D. Dixon, Polynomials with nontrivial relations between their roots, Acta Arith. 82 no. 3 (1997), 293–302. DOI MR Zbl
  • M. Drmota and M. Skałba, On multiplicative and linear independence of polynomial roots, in Contributions to general algebra 7 (Vienna, 1990),...
  • M. Drmota and M. Skałba, Relations between polynomial roots, Acta Arith. 71 no. 1 (1995), 65–77. DOI MR Zbl
  • P. Drungilas and A. Dubickas, On degrees of three algebraic numbers with zero sum or unit product, Colloq. Math. 143 no. 2 (2016), 159–167....
  • A. Dubickas, On the degree of a linear form in conjugates of an algebraic number, Illinois J. Math. 46 no. 2 (2002), 571–585. MR Zbl Available...
  • A. Dubickas, Additive relations with conjugate algebraic numbers, Acta Arith. 107 no. 1 (2003), 35–43. DOI MR Zbl
  • A. Dubickas, Multiplicative relations with conjugate algebraic numbers, Ukraïn. Mat. Zh. 59 no. 7 (2007), 890–900, and in Ukrainian Math....
  • A. Dubickas and J. Jankauskas, Simple linear relations between conjugate algebraic numbers of low degree, J. Ramanujan Math. Soc. 30 no. 2...
  • A. Dubickas and C. J. Smyth, Polynomials with three distinct zeros summing to zero (problem 11123), Amer. Math. Monthly 113 no. 10 (2006),...
  • K. Girstmair, Linear dependence of zeros of polynomials and construction of primitive elements, Manuscripta Math. 39 no. 1 (1982), 81–97....
  • K. Girstmair, Linear relations between roots of polynomials, Acta Arith. 89 no. 1 (1999), 53–96. DOI MR Zbl
  • K. Girstmair, The Galois relation x1=x2+x3 and Fermat over finite fields, Acta Arith. 124 no. 4 (2006), 357–370. DOI MR Zbl
  • K. Girstmair, The Galois relation x1=x2+x3 for finite simple groups, Acta Arith. 127 no. 3 (2007), 301–303. DOI MR Zbl
  • K. Girstmair, The Galois relations x1=x2+x3 and x1=x2x3 for certain solvable groups, Ann. Sci. Math. Québec 32 no. 2 (2008), 171–174....
  • Y. Kitaoka, Notes on the distribution of roots modulo a prime of a polynomial, Unif. Distrib. Theory 12 no. 2 (2017), 91–117. MR Zbl
  • V. A. Kurbatov, Galois extensions of prime degree and their primitive elements, Izv. Vysš. Učebn. Zaved. Matematika no. 1(176) (1977), 61–66,...
  • F. Lalande, Relations linéaires entre les racines d'un polynôme et anneaux de Schur, Ann. Sci. Math. Québec 27 no. 2 (2003), 169–175....
  • F. Lalande, La relation linéaire a=b+c+⋯+t entre les racines d'un polynôme, J. Théor. Nombres Bordeaux 19 no. 2 (2007),...
  • F. Lalande, À propos de la relation galoisienne x1=x2+x3, J. Théor. Nombres Bordeaux 22 no. 3 (2010), 661–673. DOI MR ZblC. J....
  • C. J. Smyth, Additive and multiplicative relations connecting conjugate algebraic numbers, J. Number Theory 23 no. 2 (1986), 243–254. DOI...
  • P. Virbalas, Linear relations between three conjugate algebraic numbers of low degree, J. Korean Math. Soc. 62 no. 2 (2025), 253–284. DOI...
  • T. Zheng, Characterizing triviality of the exponent lattice of a polynomial through Galois and Galois-like groups, in Computer algebra in...
  • T. Zheng, A fast algorithm for computing multiplicative relations between the roots of a generic polynomial, J. Symbolic Comput. 104 (2021),...