Reducibility of Polynomialsf(x, y) Modulo p

✍ Scribed by Wolfgang M Ruppert

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 109 KB
• Volume: 77
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2381

✦ Synopsis

We consider absolutely irreducible polynomials f # Z[x, y] with deg x f=m, deg y f=n, and height H. We show that for any prime p with p>[m(n+1) n 2 + (m+1)(n&1) m 2 ] mn+(n&1)Â2 } H 2mn+n&1 the reduction f mod p is also absolutely irreducible. Furthermore if the Bouniakowsky conjecture is true we show that there are infinitely many absolutely irreducible polynomials f # Z[x, y] which are reducible mod p where p is a prime with p H 2m .