Definable henselian valuation rings

We give model theoretic criteria for $\exists \forall$ and $\forall \exists$- formulas in the ring language to define uniformly the valuation rings $\mathcal{O}$ of models $(K, \mathcal{O})$ of an elementary theory $\Sigma$ of henselian valued fields. As one of the applications we obtain the existence of an $\exists \forall$-formula defining uniformly the valuation rings $\mathcal{O}$ of valued henselian fields $(K, \mathcal{O})$ whose residue class field $k$ is finite, pseudo-finite, or hilbertian. We also obtain $\forall \exists$-formulas $\varphi_2$ and $\varphi_4$ such that $\varphi_2$ defines uniformly $k[[t]]$ in $k((t))$ whenever $k$ is finite or the function field of a real or complex curve, and $\varphi_4$ does the job if $k$ is any number field

Uniform definability of henselian valuation rings in the Macintyre language

We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists\forall$-$\emptyset$-definable in the ring language, but not uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$. We apply these results to local fields $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$, as well as to higher dimensional local fields

Brief of Amici Curiae Public Justice, the Prisoners’ Rights Project of the Legal Aid Society of the City of New York, and the Pennsylvania Institutional Law Project in Support of Plaintiffs-Appellees (Argueta v. United States Immigration and Customs Enforcement)

Public Justice is a national public interest law firm dedicated to preserving access to justice, remedying government and corporate wrongdoing, and holding the powerful accountable in courts. As part of its access-to-justice work, Public Justice created an Iqbal Project in 2009 to combat misuse of the Supreme Court’s decision in Ashcroft v. Iqbal, 129 S. Ct. 1937 (2009). The Project tracks developments in the case law and provides assistance to counsel facing Iqbal-based motions. Public Justice is concerned that overbroad readings of Iqbal threaten to deny justice to many injured plaintiffs with meritorious claims. In addition to Public Justice’s Iqbal-related interest in this case, the firm also represents prisoners, arrestees, other detainees, their family members, and other plaintiffs in a variety of cases involving constitutional claims. See, e.g., Hui v. Castaneda, 130 S. Ct. 1845 (2010); Dillon v. Rogers, 596 F.3d 260 (5th Cir. 2010); Menotti v. City of Seattle, 409 F.3d 1113 (9th Cir. 2005); Everett v. Cherry, No. 08-00622 (E.D. Va.) (case pending). Public Justice is concerned that Appellants’ arguments regarding supervisory liability will, if accepted, prevent many plaintiffs with constitutional claims from obtaining a full remedy. The Legal Aid Society of the City of New York is a private organization that has provided free legal assistance to indigent persons in New York City for over 125 years. Through its Prisoners’ Rights Project, the Society seeks to ensure that 2 prisoners’ constitutional and statutory rights are protected. The Society advocates on behalf of prisoners in the New York City jails and New York state prisons, and conducts litigation on prison conditions. The Society often litigates claims of supervisory liability. The Pennsylvania Institutional Law Project is a private not-for-profit organization created to ensure equal access to justice for indigent institutionalized persons. Part of the Pennsylvania Legal Aid Network, the Institutional Law Project provides direct representation services, self-help and other legal materials, and class representation to eligible low-income residents of Pennsylvania’s prisons, jails, state hospitals, and state centers. The Project also takes part in advocacy and legislative initiatives concerning institutional reform in Pennsylvania

Formally Real Fields

Summary We extend the algebraic theory of ordered fields [7, 6] in Mizar [1, 2, 3]: we show that every preordering can be extended into an ordering, i.e. that formally real and ordered fields coincide.We further prove some characterizations of formally real fields, in particular the one by Artin and Schreier using sums of squares [4]. In the second part of the article we define absolute values and the square root function [5].Institute of Informatics, Faculty of Mathematics, Physics and Informatics, University of Gdansk Wita Stwosza 57, 80-308 Gdansk, PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-8 17.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Infor mation Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363-371, 2016. doi: 10.15439/2016F520.Nathan Jacobson. Lecture Notes in Abstract Algebra, III. Theory of Fields and Galois Theory. Springer-Verlag, 1964.Manfred Knebusch and Claus Scheiderer. Einf¨uhrung in die reelle Algebra. Vieweg-Verlag, 1989.Alexander Prestel. Lectures on Formally Real Fields. Springer-Verlag, 1984.Knut Radbruch. Geordnete K¨orper. Lecture Notes, University of Kaiserslautern, Germany, 1991.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Christoph Schwarzweller. Ordered rings and fields. Formalized Mathematics, 25(1):63-72, 2017. doi: 10.1515/forma-2017-0006.Christoph Schwarzweller. On roots of polynomials and algebraically closed fields. Formalized Mathematics, 25(3):185-195, 2017. doi: 10.1515/forma-2017-0018.Christoph Schwarzweller and Artur Korniłowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333-349, 2015. doi: 10.1515/forma-2015-0027.25424925

Representation of real commutative rings

During the last 10 years there have been several results on the representation of real polynomials, positive on some semi-algebraic subset of R n. These results started with a solution of the moment problem by Schmüdgen for corresponding sets. Later Wörmann realized that the same results could be obtained by the so-called “Kadison-Dubois ” Representation Theorem. The aim of our talk is to present this representation theorem together with its history, and to discuss its implication to the representation of positive polynomials. Also recent improvements of both topics by T. Jacobi and the author will be included. 1 The real representation theorem. Roughly speaking, the purpose of a representation theorem is, to start with awell-known mathematical structure, generalize (axiomatize) it and try to ‘represent ’ the generalized structures in terms of the old one. In our case, the ring that is assumed to be well-known is the ring C(X, R) of continuous real-valued functions on a compact hausdorff space X. This ring carries a natural partial order given by Thus the set f ≤ g iff f(x) ≤ g(x) for all x ∈ X. T0 = {f ∈ C(X, R) | f ≥ 0onX} is a preordering of the ring C = C(X, R), i.e., we hav