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This paper builds and extends on the authors previous work related to the algorithmic tool, Cylindrical Algebraic Decomposition (CAD), and one of its core applications, Real Quantifier Elimination (QE). These topics are at the heart of symbolic computation and were first implemented in computer algebra systems decades ago, but have recently received renewed interest as part of the ongoing development of SMT solvers for non-linear real arithmetic. First, we consider the use of iterated univariate resultants in traditional CAD, and how this leads to inefficiencies, especially in the case of an input with multiple equational constraints. We reproduce the workshop paper [Davenport \& England, 2023], adding important clarifications to our suggestions first made there to make use of multivariate resultants in the projection phase of CAD. We then consider an alternative approach to this problem first documented in [McCallum \& Brown, 2009] which redefines the actual object under construction, albeit only in the case of two equational constraints. We correct an important typo and provide a missing proof in that paper. We finish by revising the topic of how to deal with SMT or Real QE problems expressed using rational functions (as opposed to the usual polynomial ones) noting that these are often found in industrial applications. We revisit a proposal made in [Uncu, Davenport and England, 2023] for doing this in the case of satisfiability, explaining why such an approach does not trivially extend to more complicated quantification structure and giving a suitable alternative., Comment: To be submitted to Mathematics in Computer Science

Economic theory is a mathematically rich field in which there are opportunities for the formal analysis of singularities and catastrophes. This article looks at the historical context of singularities through the work of two eminent Frenchmen around the late 1960s and 1970s. Ren\'e Thom (1923-2002) was an acclaimed mathematician having received the Fields Medal in 1958, whereas G\'erard Debreu (1921-2004) would receive the Nobel Prize in economics in 1983. Both were highly influential within their fields and given the fundamental nature of their work, the potential for cross-fertilisation would seem to be quite promising. This was not to be the case: Debreu knew of Thom's work and cited it in the analysis of his own work, but despite this and other applied mathematicians taking catastrophe theory to economics, the theory never achieved a lasting following and relatively few results were published. This article reviews Debreu's analysis of the so called ${\it regular}$ and ${\it crtitical}$ economies in order to draw some insights into the economic perspective of singularities before moving to how singularities arise naturally in the Nash equilibria of game theory. Finally a modern treatment of stochastic game theory is covered through recent work on the quantal response equilibrium. In this view the Nash equilibrium is to the quantal response equilibrium what deterministic catastrophe theory is to stochastic catastrophe theory, with some caveats regarding when this analogy breaks down discussed at the end., Comment: 23 pages, 5 figures

In 1994 Lazard proposed an improved method for cylindrical algebraic decomposition (CAD). The method comprised a simplified projection operation together with a generalized cell lifting (that is, stack construction) technique. For the proof of the method's validity Lazard introduced a new notion of valuation of a multivariate polynomial at a point. However a gap in one of the key supporting results for his proof was subsequently noticed. In the present paper we provide a complete validity proof of Lazard's method. Our proof is based on the classical parametrized version of Puiseux's theorem and basic properties of Lazard's valuation. This result is significant because Lazard's method can be applied to any finite family of polynomials, without any assumption on the system of coordinates. It therefore has wider applicability and may be more efficient than other projection and lifting schemes for CAD., Comment: 21 pages

In 1990 Lazard proposed an improved projection operation for cylindrical algebraic decomposition (CAD). For the proof he introduced a certain notion of valuation of a multivariate Puiseux series at a point. However a gap in one of the key supporting results for the improved projection was subsequently noticed. In this report we study a more limited but rigorous concept of Lazard's valuation: namely, we study Lazard's valuation of a multivariate polynomial at a point. We prove some basic properties of the limited Lazard valuation and identify some relationships between valuation-invariance and order-invariance.

CT and MRI are the two most widely used imaging modalities in healthcare, each with its own merits and drawbacks. Combining these techniques in one machine could provide unprecedented resolution and sensitivity in a single scan, and serve as an ideal platform to explore physical coupling of x-ray excitation and magnetic resonance. Molecular probes such as functionalized nanophosphors present an opportunity to demonstrate a synergy between these modalities. However, a simultaneous CT-MRI scanner does not exist at this moment. As a pilot study, here we propose a mechanism in which water solutions containing LiGa5O8:Cr3+ nanophosphors can be excited with x-rays to store energy, and these excited particles may subsequently influence the T2 relaxation times of the solutions so that a difference in T2 can be measured by MRI before and after x-ray excitation. The trends seen in our study suggest that a measurable effect may exist from x-ray excitation of the nanophosphors. However, there are several experimental conditions that hinder the clarity of the results to be statistically significant up to a commonly accepted level (p=0.05), including insoluble nanoparticles and inter-scan variability. Nevertheless, the initial results from our experiments seem a consistent and inspiring story that x-rays modify MRI T2 values around nanophosphors. Upon availability of soluble nanophosphors, we will repeat our experiments to confirm these observations., Comment: 24 total pages, 9 pages of text, 10 figures with captions (6 pages), supplemental material including analysis code and tutorial (9 pages)

When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD). In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint. The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions., Comment: 40 pages

This article makes the key observation that when using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is not always the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This motivates our definition of a Truth Table Invariant CAD (TTICAD). We generalise the theory of equational constraints to design an algorithm which will efficiently construct a TTICAD for a wide class of problems, producing stronger results than when using equational constraints alone. The algorithm is implemented fully in Maple and we present promising results from experimentation., Comment: To appear in the proceedings of the 38th International Symposium on Symbolic and Algebraic Computation (ISSAC '13)

The only known constructive factorization algorithm for linear partial differential operators (LPDOs) is Beals-Kartashova (BK) factorization \cite{bk2005}. One of the most interesting features of BK-factorization: at the beginning all the first-order factors are constructed and afterwards the factorization condition(s) should be checked. This leads to the important application area - namely, numerical simulations which could be simplified substantially if instead of computation with one LPDE of order $n$ we will be able to proceed computations with $n$ LPDEs all of order 1. In numerical simulations it is not necessary to fulfill factorization conditions exactly but with some given accuracy, which we call approximate factorization. The idea of the present paper is to look into the feasibility of solving problems of this kind using quantifier elinination by cylindrical algebraic decomposition.