Equations involving the modular 𝑗-function and its derivatives

We show that, for any polynomial <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>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> F(X,Y₀,Y₁,Y₂)[X,Y₀,Y₁,Y₂] , the equation <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>z</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:msup> <m:mi>j</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:msup> <m:mi>j</m:mi> <m:mo>′′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> F(z,j(z),j^{}(z),j^{}(z))=0 has a Zariski dense set of solutions in the hypersurface <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>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> F(X,Y₀,Y₁,Y₂)=0 , unless 𝐹 is in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">C</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:math> C[X] or it is divisible by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Y</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> Y₀ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Y</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>−</m:mo> <m:mn>1728</m:mn> </m:mrow> </m:math> Y₀-1728 , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Y</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> Y₁ . Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions.