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Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple $$ (f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...

Here are two ways to define Hilbert's tenth problem over a ring $R$: Given a polynomial $p \in \mathbb Z[x_1,\ldots,x_n]$, can one decide whether it has a solution in $R^n$? Given a polynomial $p \in ...

Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers. More concretely, is there an algorithm to find $...