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Patterns and periodicity in a family of resultants

Kevin G. HareDavid McKinnonChristopher D. Sinclair — 2009

Journal de Théorie des Nombres de Bordeaux

Given a monic degree N polynomial f ( x ) [ x ] and a non-negative integer , we may form a new monic degree N polynomial f ( x ) [ x ] by raising each root of f to the th power. We generalize a lemma of Dobrowolski to show that if m < n and p is prime then p N ( m + 1 ) divides the resultant of f p m and f p n . We then consider the function ( j , k ) Res ( f j , f k ) mod p m . We show that for fixed p and m that this function is periodic in both j and k , and exhibits high levels of symmetry. Some discussion of its structure as a union of lattices is also given.

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