Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Integral decomposition of hermitian forms.

Abstract: Let K be a field of characteristic not 2 with a non-trivial involution. It is well-known that every hermitian space V over K has a decomposition into an orthogonal sum of lines. Now suppose further that K is the quotient field of a Dedekind domain ?, and that L is an ?3-lattice on the hermitian K-space V. What can be said about the decomposability of L? If K is a local field and ? its ring of integers, then it is not difficult to show that every ?-lattice splits into 1and 2-dimensional components (cf. Jacobowitz [6] and Johnson [8] for details and applications). In this paper we consider the global theory of hermitian forms, and onr main objective is to prove the following analogue of the above local result: Let K be an algebraic number field with a non-trivial involution, and let D be the ring of integers of K. Suppose V is a regular indefinite herinitian space over K of dimension n ? 5. Then every Z-lattice L on V has a splitting L = L1 1 I Lt, with rank Li ?4 for l?i?t. The decomposition of quadratic forms was investigated in [13] and [4]. In [13] Watson showed that every indefinite quadratic form over Q of rank n ? 12 has a decomposition over Z. (Watson has also shown the number 12 to be best possible.) In [4] the present author extended Watson's result to quadratic forms over the flasse domains of global fields. In particular it was shown that given any algebraic number field K with ring of integers ?,7 there exists a natural number n (S) suich that every ?-lattice L with rank L ? n(s) splits non-trivially if its underlying quadratic space is indefinaite. But it was also seen that n (O) depends on the choice of ?) and ma;y be arbitrarily large. More explicitly, given any m > 0, there exist K and i such that there is an indecomposable ?-lattice on some indefinite quadratic Kspace of dimension greater than m. Thus our present result for hermitian forms is much stronger than that for quadratic forms. With regard to computation, herimitian forms are more manageable than quadratic forms, chiefly because dimension and discriminant suffice as local fractional invariants (the

Cyclic covers of the projective line, their jacobians and endomorphisms

Abstract: We study the endomorphism ring $End(J(C))$ of the complex jacobian $J(C)$ of a curve $y^p=f(x)$ where $p$ is an odd prime and $f(x)$ is a polynomial with complex coefficiens of degree $n>4$ and without multiple roots. Assume that all the coefficients of $f$ lie in a (sub)field $K$ and the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$. Then we prove that $End(J(C))$ is the ring of integers in the in the $p$th cyclotomic field, if $p$ is a Fermat prime (e.g., $p=3,5,17,257$). Similar results for $p=2$ (the case of hyperelliptic curves) were obtained by the author in Math. Res. Lett. 7(2000), 123--132.

Skeletons and tropicalizations

Abstract: Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ in the analytification of $X$. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker--Payne--Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on $S(\mathscr X, H)$. For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels--Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.

CTRU, a polynomial analogue of NTRU

Abstract: CTRU, a new public-key cryptosystem is introduced. In this analogue of NTRU, the ring of integers is replaced by the ring of polynomials in one variable over a finite field. Attacks based on either the LLL algorithm or the Chinese Remainder Theorem are avoided. An important tool of cryptanalys- is is the Popov normal form of matrices with polynomial entries. The speed of encryption/decryption of CTRU is the same as NTRU for the same value of N. An implementation in Aldor is described.