Most of the documentation provided for Mathematica is concerned with explaining what Mathematica does, not how it does it. But the purpose of this is to say at least a little ...

InterpolateRoot[lhs == rhs, {x, x_0, x_1}] searches for a numerical solution to the equation lhs == rhs using x_0 and x_1 as the first two values of x.

AppendColumns, AppendRows, and BlockMatrix are available using the Mathematica kernel functions Join and ArrayFlatten. TakeRows, TakeColumns, TakeMatrix, and SubMatrix are ...

CreateWindow[] creates an empty window in the front end.CreateWindow[nb] creates a window displaying the notebook expression nb, and opens it in the front ...

Element[x, dom] or x \[Element] dom asserts that x is an element of the domain dom. Element[{x_1, x_2, ...}, dom] asserts that all the x_i are elements of dom. Element[patt, ...

Integers represents the domain of integers, as in x \[Element] Integers.

A Diophantine polynomial system is an expression constructed with polynomial equations and inequalities combined using logical connectives and quantifiers where the variables ...

Modulus -> n is an option that can be given in certain algebraic functions to specify that integers should be treated modulo n.

ComplexExpand[expr] expands expr assuming that all variables are real. ComplexExpand[expr, {x_1, x_2, ...}] expands expr assuming that variables matching any of the x_i are ...

TrigReduce[expr] rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.