Explicit bounds for the solutions of superelliptic equations over number fields

Let f be a polynomial with coefficients in the ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>O</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> {OS} of S -integers of a number field K , b a non-zero S -integer, and m an integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {≥ 2} . We consider the following equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo>⋆</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {()} : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>y</m:mi> <m:mi>m</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {f(x)=byᵐ} in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi>O</m:mi> <m:mi>S</m:mi> </m:msub> </m:mrow> </m:math> {x,y∈ OS} . Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> <m:mo>,</m:mo> <m:mi>m</m:mi> </m:mrow> </m:math> {K,S,f,m} and the S -norm of b for the heights of the solutions x of equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo>⋆</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {()} . Further, we give an explicit bound C in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> {K,S,f} and the S -norm of b such that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>></m:mo> <m:mi>C</m:mi> </m:mrow> </m:math> {m>C} equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo>⋆</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {()} has only solutions with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>y</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {y=0} or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of Bérczes, Evertse, and Győry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the S -norm of b instead of its height.