An official website of the United States government
We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.
more »
« less
Theories with few non-algebraic types over models, and their decompositions
We consider several ways of decomposing models into parts of bounded size forming a congruence over a base, and show that admitting any such decomposition is equivalent to mutual algebraicity at the level of theories. We also show that a theory T T is mutually algebraic if and only if there is a uniform bound on the number of coordinate-wise non-algebraic types over every model, regardless of its cardinality.
more »
« less
- Award ID(s):
- 1855789
- PAR ID:
- 10409419
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 150
- Issue:
- 759
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 4021 to 4026
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We continue the study of n -dependent groups, fields and related structures, largely motivated by the conjecture that every n -dependent field is dependent. We provide evidence toward this conjecture by showing that every infinite n -dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in n -dependent groups, generalizing Shelah’s absoluteness of $$G^{00}$$ in dependent theories and relative absoluteness of $$G^{00}$$ in $$2$$ -dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $$2$$ -dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are $$2$$ -dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that n -dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $$2$$ -dependence and use it to deduce $$2$$ -dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory T by a generic predicate is dependent if and only if it is n -dependent for some n , if and only if the algebraic closure in T is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.more » « less
-
null (Ed.)The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over ℂ((t)), and show that the Ceresa class in each of these settings is torsion.more » « less
-
Amir Hashemi (Ed.)We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, require argument/value triples (w^i, f(w^i), f'(w^i)) for i=0,...,t + ceiling( (t+1)/2 ) - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B >= t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, ceiling( t/2 ) + B triples, but then never return an incorrect output. The algorithms are based on Prony's sparse interpolation algorithm. While Prony's algorithm and its variants use fewer values, namely, 2t+1 and t+B values f(w^i), respectively, they need more arguments w^i. The situation mirrors that in algebraic error correcting codes, where the Reed-Solomon code requires fewer values than the multiplicity code, which is based on Hermite interpolation, but the Reed-Solomon code requires more distinct arguments. Our sparse Hermite interpolation algorithms can interpolate polynomials over finite fields and over the complex numbers, and from floating point data. Our Prony-based approach does not encounter the Birkhoff phenomenon of Hermite interpolation, when a gap in the derivative values causes multiple interpolants. We can interpolate from t+1 values of f and 2t-1 values of f'.more » « less
-
We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/15956
