Ricardo Jesús Palomino Piepenborn

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Preprints

1. Model theory of local real closed SV-rings of finite rank. PDF. ArXiv. Expand abstract. (to be submitted)

   This note begins the model-theoretic study of local real closed SV-rings of finite rank; to this end, a structure theorem for reduced local SV-rings of finite rank is given and branching ideals in local real closed rings of finite rank are analysed. The class of local real closed SV-rings of rank n ∈ ℕ^≥2 is elementary in the language of rings ℒ := {+,−,·,0,1} and its ℒ-theory T_n has a model companion T_n,1; models of T_n,1 are n-fold fibre products (((V₁× _k V₂)× _k V₃) ...× _k V_n−1)× _k V_n of non-trivial real closed valuation rings V_i with isomorphic residue field k . The ℒ-theory T_n,1 is complete, decidable, and NIP. After enriching ℒ with a predicate for the maximal ideal, models of T_n have prime extensions in models of T_n,1, and T_n,1 is the model completion of T_n in this enriched language. A quantifier elimination result for T_n,1 is also given. The class of those local real closed SV-rings of rank n ∈ ℕ^≥2 which are n-fold fibre products (((V₁×_WV)×_WV₃) ...×_WV_n-1)×_WV_n of non-trivial real closed valuation rings V_i along surjective morphisms V_i↠W onto a non-trivial domain W is elementary in the language of rings, and its ℒ-theory T_n,2 is also complete, decidable, and NIP; after enriching ℒ with predicates for the maximal ideal and the unique branching ideal, T_n,2 is model complete.

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Work in progress

1. (with Marcus Tressl) Cross-sections of real closed rings.

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Unpublished notes

1. Cross-sections of divisible abelian o-groups via tame pairs. PDF. Expand abstract.

   In this note several equivalent characterizations are given for a divisible subgroup ∆′ ⊆ Γ of a divisible group Γ to be the image of a section of a given surjective o-group homomorphism f : Γ ↠ ∆ using the order-theoretic notion of tameness (equivalently, relative Dedekind completeness). The note concludes with an application of these characterizations to real closed valued fields.

2. (LMS-funded 2019 summer project) Invariants of the simultaneous conjugacy problem for matrices over ℂ. PDF. Expand abstract.

   In this paper we establish a way of checking if two d-tuples of n × n matrices over ℂ are conjugate using a function F_d: ℂ^dn2 → ℂ^k that outputs the coefficients in the polynomials of the reduced Gröbner basis of certain ideal. We give bounds for the k ∈ ℕ in the codomain of F_d and we illustrate the solution of the problem by means of small examples implemented in Singular.

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Slides of selected talks

1. (OAL 2025) Local real closed SV-rings of finite rank and their model theory. PDF. Expand abstract.

   A commutative unital ring is an SV-ring if all its residue domains (that is, all its quotients by prime ideals) are valuation rings. SV-rings were first introduced in the context of rings of continuous functions C(X), and later studied in the more general context of f-rings. Among all f-rings, Schwartz’s real closed rings form a subclass which play a central role in semi-algebraic geometry; examples of such rings are the rings C(X) and rings of continuous semi-algebraic functions on a semi-algebraic set over a real closed field. Residue domains of real closed SV-rings are real closed valuation rings (equivalently, convex subrings of real closed fields), and their model theory is well understood. The goal of this talk is to present an approach to the model-theoretic analysis of some classes of local real closed SV-rings in terms of their residue domains. A local real closed SV-ring has rank n if it has exactly n minimal prime ideals, and the class of local real closed SV-rings of rank n is elementary in the language of rings; a geometric example of such a ring is the ring of germs G of continuous semi-algebraic functions at a point in a semi-algebraic curve with exactly n half-branches. First, I will spell out a representation theorem for local real closed SV-rings of rank n, which describes them as iterated fibre products of n non-trivial real closed valuation rings (that is, those real closed valuation rings which are not fields). I will then focus on the class of n-fold fibre products of non-trivial real closed valuation rings with isomorphic residue field along their residue field map; an example of such a ring is the ring of germs G. I will sketch proofs of the main model-theoretic properties of this class, in particular: 1. It has a recursive axiomatization in the language of rings and its theory is complete (hence decidable); and 2. Its theory is model complete and it is the model companion of the theory of local real closed (SV-) rings of rank n. The same results as in 1) hold for another class of n-fold fibre products of non-trivial real closed valuation rings. As a consequence, the theory of local real closed SV-rings of rank 2 has exactly two completions, both of which are decidable. I will conclude by showing how these results suggest an elementary classification of all local real closed SV-rings of finite rank in terms of the isomorphism type of their poset of branching prime ideals.

2. (OAL-RAG 2024) Relative quantifier elimination for the lattice-ordered module of continuous semi-algebraic functions on a curve. PDF. Expand abstract.

   In the late 1980s, Shen and Weispfenning proved, via relative quantifier elimination in a suitable 2-sorted language, that under a mild condition on a divisible abelian lattice-ordered group G of functions, the theory of G is completely determined by the theory of its lattice of zero sets. In this talk I will give the relevant context and details of their result, to then explain how the ideas in their proof can be adapted to lattice-ordered modules M of continuous semi-algebraic functions on a curve by enriching their 2-sorted language with a new sort for a real closed valuation ring; as a consequence of the method, decidability of M is obtained whenever the base field is a recursive real closed field.

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Teaching (university level)

Graduate teaching assistant at the University of Manchester: 2019 - 2024

• Tutorials (∼50 students):
  • 2022: Sequences and series.
  • 2021: Introduction to mathematical logic.
  • 2020: Mathematical logic, Mathematical workshop.
  • 2019: Combinatorics and graph theory.
• Supervisions (∼15 students; leading problem sessions and weekly marking):
• 2022: Calculus and applications.
• 2021: Foundations of pure mathematics.
• 2020: Linear algebra.
• 2019: Foundations of pure mathematics.
• Coursework marking:
• 2024: Algebraic structures 2.
• 2021: Foundations of pure mathematics.
• 2020: Mathematical workshop.

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Event Organization

• Conferences:
• 2022 Manchester Mathematics Research Students’ Conference (co-organiser; ∼70 attendees).
• 2022 British Postgraduate Model Theory Conference (co-organiser; ∼50 attendees).
• Seminars (at the University of Manchester):
• 2021/2022 Pure Postgraduate Seminar (co-organiser; ∼ 20 attendees).
• 2021/2022 Student Logic Seminar (organiser; ∼ 10 attendees).
• 2019/2020 MSc Pure Mathematics Seminar (organiser; ∼ 10 attendees).

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Outreach

1. Mathematical Modelling and Problem Solving Day: Cryptography. PDF. Show summary.

   Yearly one-day event organized by the University of Manchester aimed at Year 12 students for them to take part in real world problem solving projects lead by PhD students. I lead the cryptography project for the 2021 and 2022 events; this consisted in explaining some of the early history of cryptography, and then going through some basic modular arithmetic to use it in a particular instance of the Diffie-Hellman key exchange protocol.

2. Mathematics at University and Infinite Sets. PDF. Show summary.

   Talk given at Trinity Church of England High School in Manchester. I introduced university-level mathematics by first highlighting key differences with high school mathematics, and then working through a formalization of the concept of infinite set illustrated and motivated by Hilbert's hotel.

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