Algebraic series and valuation rings over nonclosed fields

Suppose that k is an arbitrary field. Let k[[x₁,…,x_n]] be the ring of formal power series in n variables with coefficients in k. Let be the algebraic closure of k and . We give a simple necessary and sufficient condition for σ to be algebraic over the quotient field of k[[x₁,…,x_n]]. We also characterize valuation rings V dominating an excellent Noetherian local domain R of dimension 2, and such that the rank increases after passing to the completion of a birational extension of R. This is a generalization of the characterization given by M. Spivakovsky [Valuations in function fields of surfaces, Amer. J. Math. 112 (1990) 107–156] in the case when the residue field of R is algebraically closed.     

Algebras over infinite fields,revisited

We generalize an earlier result of the first author by showing that under “the usual cardinality-dimension hypothesis” an algebra in which nonzero divisors are invertible is algebraic. This result is then used to show that affine FBN rings over an uncountable algebraically closed field are PI.     

The Picard group of algebras over an algebraically nonclosed field

Let k be a field that is not algebraically closed, and let A be a k-algebra, whose each maximal ideal has residue field k. We prove that each element of the Picard group of A is of finite order and give an optimal upper bound for its order.     

The model theory of differential fields revisited

The intent of this article is to provide a general and elementary account of the model theory of differential fields, collecting together various results (many without proof) and offering a few algebraic details for the logician reader. The first model-theoretic look at differential fields was taken by Abraham Robinson in the context of model completeness, while later developments have served to illustrate concepts developed by Morley and Shelah. Preparation of this paper supported in part by N.S.F. Grant MPS 75-08241.     

The genus of maximal function fields over finite fields

We prove that if there exists a maximal function field of one variable of genusg over $\mathbb{F}_{q^2 } $ , theng≤(q?1) ²/4 org=qr/2 withq?1/2≤r≤q?1.     

Uniqueness of exceptional singular quartics

We prove that given a general collection of 14 points of ( an infinite field) there is a unique quartic hypersurface that is singular on .

This completes the solution to the open problem of the dimension of a linear system of hypersurfaces of that are singular on a collection of general points.

     

The uncertainty principle over finite fields

In this paper we study the uncertainty principle (UP) connecting a function over a finite field and its Mattson-Solomon polynomial, which is a kind of Fourier transform in positive characteristic. Three versions of the UP over finite fields are studied, in connection with the asymptotic theory of cyclic codes. We first show that no finite field satisfies the strong version of UP, introduced recently by Evra, Kowalsky, Lubotzky, 2017. A refinement of the weak version is given, by using the asymptotic Plotkin bound. A naive version, which is the direct analogue over finite fields of the Donoho-Stark bound over the complex numbers, is proved by using the BCH bound. It is strong enough to show that there exist sequences of cyclic codes of length n, arbitrary rate, and minimum distance $\mathrm{\Omega }(n^{\alpha })$ for all $0<\alpha <1/2$. Finally, a connection with Ramsey Theory is pointed out.     

The fermat equation over quadratic fields

Kummer's method of proof is applied to the Fermat equation over quadratic fields. The concept of an m-regular prime, p, is introduced and it is shown that for certain values of m, the Fermat equation with exponent p has no nontrivial solutions over the field Q(√m).     

The u-invariant of one-dimensional function fields over real power series fields

The u-invariant of a field K of characteristic not 2 is the largest dimension of an anisotropic torsion quadratic form over K. We extend the recent theorem by Harbater, Hartmann, and Krashen on the u-invariant of a one-dimensional function field over a complete discretely valued field to the case where the residue field can be ordered.     

Fano surfaces of real quartics

We prove certain properties of the Fano surface of bitangents to the real quartic in projective 3-space. In particular, we estimate the dimension of the cohomology of the real part of this surface in terms of the dimension of the cohomology of the real part of the quartic, and compute its Euler characteristic.     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

Patching over fields

We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.     

The k-subset sum problem over finite fields

The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k-subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k-subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results.