It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi-)orthogonal rational functions and linear pencils zB − A of two tridiagonal matrices A, B, following Spiridonov and Zhedanov. In the present paper, beside revisiting the underlying generalized Favard t...

Given A spline curves and A patch surfaces that are implicitly de ned on triangles and tetrahedra we determine their NURBS representations We provide a trimmed NURBS form for A spline curves and a parametric tensor product NURBS form for A patch surfaces We concentrate on cubic A patches providing a C continuous surface that interpolates a given triangulation together with surface normals at th...

We give a variant of Barvinok’s algorithm for computing a short rational generating function for the integer points in P := {x ∈ Rn : Ax ≤ b}; a use of which is to count the number of integer points in P . We use an algebraic perturbation, replacing each bi with bi + τ i, where τ > 0 is an arbitrarily small indeterminate. Hence, our new right-hand vector has components in the ordered ring Q[τ ]...

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.

Abstract The extended coset leader weight enumerator of the generalized Reed–Solomon $$[q+1,q-3,5]_q$$ [ q + 1 , - 3 5 ] code is...

Given a smooth cubic hypersurface X over finite field of characteristic greater than 3 and two generic points on X, we use function analogue the Hardy–Littlewood circle method to obtain an asymptotic formula for number degree d k-rational curves passing through those points. We this deduce dimension irreducibility moduli space parametrising such curves, large enough d.

Let X ⊂ P be a geometrically integral cubic hypersurface defined over Q, with singular locus of dimension 6 dimX − 4. Then the main result in this paper is a proof of the fact that X(Q) contains Oε,X(B ) points of height at most B.

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in Pn of degree 2d ≤ min(n+4, 2n−2) and dimension at least three is irreducible and of the expected dimension.