We show that recent results of Coppersmith, Boneh, Durfee and Howgrave-Graham actually apply in the more general setting of (partially) approximate common divisors. This leads us to consider the question of “fully” approximate common divisors, i.e. where both integers are only known by approximations. We explain the lattice techniques in both the partial and general cases. As an application of ...

The recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with twoand three-dimensional irreducible representations. This paper details the mathematics of the two finite simple groups with such representations, the Icosahedral group A5, a subgroup of SO(3), and PSL2(7), a subgroup of SU(3). ∗E-mail: [email protected] †E-m...

In this paper we exhibit an example of a three-dimensional regular local domain (A,n) having a height-two prime ideal P with the property that the extension P of P to the n-adic completion  of A is not integrally closed. We use a construction we have studied in earlier papers: For R = k[x, y, z], where k is a field of characteristic zero and x, y, z are indeterminates over k, the example A is...

Algebraic independence is a fundamental notion in commutative algebra that generalizes independence of linear polynomials. Polynomials {f1, . . . , fm} ⊂ K[x1, . . . , xn] (over a field K) are called algebraically independent if there is no non-zero polynomial F such that F (f1, . . . , fm) = 0. The transcendence degree, trdeg{f1, . . . , fm}, is the maximal number r of algebraically independen...

For a radical ideal I, the symbolic power I is the collection of elements that vanish to order at least p at each point of Zeros(I). If I is actually prime, then I is the I-associated primary component of I; if I is only radical, writing I = C1 ∩ · · · ∩Cs as an intersection of prime ideals, I = C (p) 1 ∩ · · · ∩ C (p) s . The inclusion I ⊆ I always holds, but the reverse inclusion holds only i...

It is known that Fairlie–Odesskii algebra U ′ q(so3) appears as algebra of observables in quantum gravity in (2 + 1)-dimensional de Sitter space with space being torus. In this paper, we study the center of this algebra at q a root of 1. It turns out that Casimir elements in this case are algebraically dependent. Using realization of the algebra U ′ q(so3) in terms of quantized lengths of geode...

1.1. Motivation. Over C and over non-archimedean fields, analytification of algebraic spaces is defined as the solution to a quotient problem. Such analytification is interesting, since in the proper case it beautifully explains the essentially algebraic nature of proper analytic spaces with “many” algebraically independent meromorphic functions. (See [A] for the complex-analytic case, and [C3]...

Testing the validity of probabilistic models containing unmeasured (hidden) variables is shown to be a hard task. We show that the task of testing whether models are structurally incompatible with the data at hand, requires an exponential number of independence evaluations, each of the form: "X is conditionally independent of Y, given Z." In contrast, a linear number of such evaluations is requ...

In this paper, we analyze the fundamental conditions for low-rank tensor completion given the separation or tensor-train (TT) rank, i.e., ranks of unfoldings. We exploit the algebraic structure of the TT decomposition to obtain the deterministic necessary and sufficient conditions on the locations of the samples to ensure finite completability. Specifically, we propose an algebraic geometric an...

It is shown that if F\ and Fi are algebraically closed fields of nonzero characteristic p and F\ is not isomorphic to a subfield of F2 , then F\ does not embed in the skew field of quotients 0Fj of the ring of morphisms of the additive group of F2 . From this fact and results of Evans and Hrushovski, it is deduced that the algebraic closure geometries G(K¡/Fi) and (7(^2/^2) are isomorphic if an...