Tagged Questions

Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. ...

415 views

Are there any integers which can't be written as a sum of two fourth powers minus a cube?

To be precise, I am asking: Does there exist an integer $k$ such that there do not exist (possibly negative) integers $x,y,z$ satisfying $x^4+y^4=z^3+k$? Heuristically the answer must be yes, in ...
488 views

218 views

When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by $$L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}.$$ A well-...
179 views

Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either returns true or false, we say that $\mathcal{A}$ works for $P$...
110 views

199 views

186 views

Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$ I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that $(x_{i},y_{i}),i=1,2,\cdots,N$...
224 views

Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...