This paper studies so-called “null polynomials modulo m”, i.e., polynomials with integer coefficients that satisfy f(x) ≡ 0 (mod m) for any integer x. The study on null polynomials is helpful to reduce congruences of higher degrees modulo m and to enumerate equivalent polynomial functions modulo m, i.e., functions over Zm = {0, · · · , m − 1} generated by integer polynomials. The most well-know...

we study connected orientable spacelike hypersurfaces $x:m^{n}rightarrowm_q^{n+1}(c)$, isometrically immersed into the riemannian or lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~l_kx=ax+b$,~ where $l_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $h_{k+1}$ of the hypersurface for a fixed integer $0leq k

Some integer factorization algorithms require several vectors in the null space of a sparse m x n matrix over the field GF(2). We modify the Lanczos algorithm to produce a sequence of orthogonal subspaces of GF(2)", each having dimension almost N, where N is the computer word size, by applying the given matrix and its transpose to N binary vectors at once. The resulting algorithm takes about n ...

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-man...

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-man...

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-man...