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Reducibility of polynomials \(f(x,y)\) modulo \(p\). (English) Zbl 0931.11005

Let \(f=\sum_{i,j}a_{ij}x^iy^j\in{\mathbb Z}[x,y]\) be an absolutely irreducible polynomial with \(\deg_x f=m\), \(\deg_y f=n\) and let \(H(f)= \max_{i,j} |a_{ij}|\). The author shows that the reduction of \(f\) modulo a prime \(p\) is still absolutely irreducible whenever \[ p>[m(n+1)n^2+(m+1)(n-1)m^2]^{mn+(n-1)/2}\cdot H(f)^{2mn+n-1}. \] This improves upon a result of U. Zannier [Arch. Math. 68, 129-138 (1997)] and can be seen as a refinement of a previous result of W. Ruppert [J. Reine Angew. Math. 369, 167-191 (1986; Zbl 0584.14012)] (in the latter case the estimate on \(p\) was given in terms of \(H(f)\) and the total degree \(d\) of the polynomial \(f\)).
Regarding the quality of the estimate, the author observes that, if the Bouniakowsky conjecture (or the better known Schinzel H conjecture) is true, then there are infinitely many polynomials \(f\in{\mathbb Z}[x,y]\) which are reducible modulo \(p\), where \(p\) is a prime with \(p\geq H(f)^{2m}\).

MSC:

11C08 Polynomials in number theory
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References:

[1] Bouniakowsky, V., Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la décomposition des entiers en facteurs, Mém. Acad. Sci. St. Pétersbourg (6) Sci. Math. Phys., 6, 305-329 (1857)
[2] Ruppert, W., Reduzibilität ebener Kurven, J. Reine Angew. Math., 369, 167-191 (1986) · Zbl 0584.14012
[3] Ruppert, W., Reducibility of polynomials \(a_0xa_1xya_2xy^2p\), Arch. Math., 72, 47-55 (1999)
[4] Schinzel, A.; Sierpinski, W., Sur certaines hypothèses concernant les nombres premiers, Acta Arith., 4, 185-208 (1958) · Zbl 0082.25802
[5] Zannier, U., On the reduction modulo \(pfxy\), Arch. Math., 68, 129-138 (1997)
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