I have seen the lecture note of Prof. Gandhi from BITS, and I could not digest to obtain (X, Y, Z) values in integers. Is there a method to analyze the nature of solutions in integers of the following in-equations. Where these kind of equations, we usually study?
$X^Y - XZ^2 - Y^X + YZ^2 \geq 0$ and $X^Y - XZ^2 + Y^X - YZ^2 \geq 0$
generalize the solutions of the above cited problem.
Lecture notes is may not be available on the web. I attended the conference and Prof. Gandhi said that, he found the equations of the above type from his own lecture note. He also explained the nature of such equations and their resolvability. But, I didn't understand. Why I am asking this question, If I know the procedure to find solutions and then I can find solutions in the case of < = 0 and I will prepare such more equations and significance in math.
may be these are Diophantine inequalities. I am not sure. I need to learn, how to find or generalize such equations? Thank you