MSC: 14M25 06D50Keywords: Q-factorial complete toric varieties Gale duality Weighted projective spaces Hermite normal form Smith normal formThe present paper is devoted to discussing Gale duality from the Z-linear algebraic point of view. This allows us to isolate the class of Q-factorial complete toric varieties whose class group is torsion free, here called poly weighted spaces (PWS). It provides an interesting generalization of weighted projective spaces (WPS).

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the p-adic numbers Qp. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of a MCF, and we perform a general study about their convergence in Qp.In particular, we derive some conditions about their convergence and we prove that convergent MCFs always strongly converge in Qp contrarily to the real case where strong convergence is not ever guaranteed. Then, we focus on a specific algorithm that, starting from a m-tuple of numbers in Qp, produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized p-adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.

No abstract

Background: Estimating the risk of developing subsequent primary tumours in a population is difficult since the occurrence probability is conditioned to the survival probability.

The present paper is devoted to generalizing, inside the class of projective toric varieties, the classification [2], performed by Batyrev in 1991 for smooth complete toric varieties, to the singular Q-factorial case.Date: September 18, 2018. 1991 Mathematics Subject Classification. 14M25; (52B20; 52B35). Key words and phrases. Q-factorial complete toric variety, projective toric bundle, secondary fan, Gale duality, fan and weight matrices, toric cover, splitting fan, primitive collection and relation.