In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x+ 1), f(x+ 2), ..., f(x+ κ)) are quadratic non-residue is ≈ 1 2 . In particular for f(x) = x + ax+ b corresponding to the elliptic curve y = x + ax+ b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly dis...

When [Formula: see text] is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory Algebra Arithmetic, Proceedings Karpacz, Poland, Lecture Notes Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that total residue map text], which picks out constant term Laurent series, definable language rings with pa...

Let X be a smooth algebraic surface over a perfect field k. Consider pairs x ∈ C , x is a closed point of X , C is either an irreducible curve on X which is smooth at x, or an irreducible analytic branch near x of an irreducible curve on X . As in the previous section 1 for every such pair x ∈ C we get a two-dimensional local field Kx,C . If X is a projective surface, then from the adelic descr...

We proceed with an analysis of 1-basedness for bounded hyperdefinable groups of the form G/G where G = E(K) is the semialgebraic connected component of the K-points of an elliptic curve over a saturated real closed field K, or G a truncation of E(K); we follow the method developed in [1]. We then relate the map G→ G/G with the algebraic geometric notion of reduction, and we characterize 1-based...

a review of damages caused by landslide shows the need for studying effective parameters in landslide occurrence and prediction. this study aims to improve landslide prediction results for tutkabon region in gilan province. to this end, shannon entropy theory was employed for modeling and considering data uncertainty. slope, height, geomorphologic conditions, earth’s curve, closeness to river, ...

Let f be a projective morphism from a smooth algebraic surface X to a smooth algebraic curve S over a perfect field k . Using the adelic language we give some relative constructions of residues and symbols and apply them to the Gysin morphism for differentials and algebraic cycles. The first section of this article is devoted to various constructions of relative residue maps from differentials ...