Diophantine equation Updated +Created
Sometimes systems of Diophantine equations are considered.
Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.
The general problem is known to be undecidable: Hilbert's tenth problem.
The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.
Hilbert's problems Updated +Created
He's a bit overly obsessed with polynomials for the taste of modern maths, but it's still fun.
Linear polynomial Updated +Created
A polynomial of degree 1, i.e. of form .
Multivariate polynomial Updated +Created
A polynomial with multiple input arguments, e.g. with two inputs and :
as opposed to a polynomial with a single argument e.g. one with just :
Polynomial over a commutative ring Updated +Created
In particular, Hilbert's tenth problem is about polynomials over the integers, which is a commutative ring, and therefore brings mindshare to this definition.
Polynomial over a ring Updated +Created
However, there is nothing in the immediate definition that prevents us from having a ring instead, i.e. a field but without the commutative property and inverse elements.
The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms:
while for a field they would all go into a single term:
so when considering a polynomial over a ring we end up with a lot more more possible terms.