Small solutions to systems of polynomial equations with integer coefficients

Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica

Volumn 19, Issue 2, 2011, Pages 89-100

  Cipu, Mihai ^a  

^a INSTITUTE OF MATHEMATICS   (Romania)

Author keywords

Hilbert's tenth problem; Quantifier elimination; Systems of polynomial equations; Waldi's determinantal inequality

Indexed keywords

EID: 79956362150     PISSN: 12241784     EISSN: 18440835     Source Type: Journal    
DOI: None     Document Type: Article

References (20)

1. 2 + cx + d, J. London Math. Soc., 43(1968), 1-9.
2. S. Basu, R. Pollack, M. F. Roy, Algorithms in real algebraic geometry, 2nd ed., Springer, Berlin, 2006.
3. D. N. Bernshtein, The number of roots of a system of equations, Anal. Appl., 9(1975), 183-185.
4. L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and real computation, Springer, New York, 1998.
5. E. Bombieri, J. Vaaler, On Siegel's lemma, Invent. Math., 7(1983), 11-32.
6. I. Borosh, M. Flahive, D. Rubin, B. Treybig, A sharp bound for solutions of linear Diophantine equations, Proc. Amer. Math. Soc., 105(1989), 844-846.
7. Y. Bugeaud, On the size of integer solutions of elliptic equations, Bull. Austral. Math. Soc., 57(1998), 199-206.
8. Y. Bugeaud, On the size of integer solutions of elliptic equations, II, Bull. Greek Math. Soc., 43(2000), 125-130.
9. M. A. Bennett, M. Cipu, M. Mignotte, R. Okazaki, On the number of solutions of simultaneous Pell equations, II, Acta Arith., 122(2006), 407-417.
10. M. Cipu, M. Mignotte, On the number of solutions to systems of Pell equations, J. Number Th., 125(2007), 356-392.
11. G. Faltings, EndlichkeitssÄatze fÄur abelsche VarietÄaten Äuber ZahlkÄorpern, Invent. Math., 73(1983), 349-366. Erratum ibidem, 75(1984), 381.
12. J. P. Jones, D. Sato, H. Wada, D. Wiens, Diophantine representation of the set of prime numbers, Amer. Math. Monthly, 83(1976), 449-464.
13. A. G. Kushnirenko, Polyμedres de Newton et nombres de Milnor, Invent. Math., 32(1976), 1-31.
14. Yu. V. Matiyasevich, Hilbert's tenth problem, MIT Press, Cambridge, MA, 1993.
15. A. Tyszka, Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients, available at the address http://arxiv.org/abs/math/0702558v84.
16. A. Tyszka, Some conjectures on addition and multiplication of complex (real) numbers, Int. Math. Forum, 4(2009), 521-530.
17. A. Tyszka, A hypothetical upper bound for the solutions (the number of solutions) of a Diophantine equation with a finite number of solutions, http://arxiv.org/0901.2093v80.
18. A. Tyszka, Two conjectures on the arithmetic in R and C, Math. Log. Quart., 56(2010), 175-184.
19. N. N. Vorobjov, Jr., Estimates of real roots of a system of algebraic equations, J. Sov. Math., 34(1986), 1754-1762.
20. R. Waldi, ÄUber ein Analogon zu Hadamards Ungleichung, Arch. Math., 59(1992), 15-20.