Detecting Simultaneous Integer Relations for Several Real Vectors

An algorithm which either finds an nonzero integer vector ${\mathbf m}$ for given $t$ real $n$-dimensional vectors ${\mathbf x}_1,...,{\mathbf x}_t$ such that ${\mathbf x}_i^T{\mathbf m}=0$ or proves that no such integer vector with norm less than a given bound exists is presented in this paper. The cost of the algorithm is at most ${\mathcal O}(n^4 + n^3 \log \lambda(X))$ exact arithmetic operations in dimension $n$ and the least Euclidean norm $\lambda(X)$ of such integer vectors. It matches the best complexity upper bound known for this problem. Experimental data show that the algorithm is better than an already existing algorithm in the literature. In application, the algorithm is used to get a complete method for finding the minimal polynomial of an unknown complex algebraic number from its approximation, which runs even faster than the corresponding \emph{Maple} built-in function.Comment: 10 page

On Number of Circles Intersected by a Line

AbstractConsider a set U of circles in the plane such that any line intersects at least one of those circles. For a given natural number m, is there a line in the plane intersecting at least m circles in U? In this paper this problem is solved. Our result is also generalized to compact convex subsets and to higher dimensional cases