This thesis develops various aspects of Algebraic Geometry and its applications in different fields of science. In Chapter 2 we characterise the feasible set of an optimisation problem relevant in chemical process engineering. We consider the polynomial dynamical system associated with mass-action kinetics of a chemical reaction network. Given an initial point, the attainable region of that point is the smallest convex and forward closed set that contains the trajectory. We show that this region is a spectrahedral shadow for a class of linear dynamical systems. As a step towards representing attainable regions we develop algorithms to compute the convex hulls of trajectories. We present an implementation of this algorithm which works in dimensions 2,3 and 4. These algorithms are based on a theory that approximates the boundary of the convex hull of curves by a family of polytopes. If the convex hull is represented as the output of our algorithms we can also check whether it is forward closed or not. Chapter 3 has two parts. In this first part, we do a case study of planar curves of degree 6. It is known that there are 64 rigid isotopy types of these curves. We construct explicit polynomial representatives with integer coefficients for each of these types using different techniques in the literature. We present an algorithm, and its implementation in software Mathematica, for determining the isotopy type of a given sextic. Using the representatives various sextics for each type were sampled. On those samples we explored the number of real bitangents, inflection points and eigenvectors. We also computed the tensor rank of the representatives by numerical methods. We show that the locus of all real lines that do not meet a given sextic is a union of up to 46 convex regions that is bounded by its dual curve. In the second part of Chapter 3 we consider a problem arising in molecular biology. In a system where molecules bind to a target molecule with multiple binding sites, cooperativity measures how the already bound molecules affect the chances of other molecules binding. We address an optimisation problem that arises while quantifying cooperativity. We compute cooperativity for the real data of molecules binding to hemoglobin and its variants. In Chapter 4, given a variety X in n-dimensional projective space we look at its image under the map that takes each point in X to its coordinate-wise r-th power. We compute the degree of the image. We also study their defining equations, particularly for hypersurfaces and linear spaces. We exhibit the set-theoretic equations of the coordinate-wise square of a linear space L of dimension k embedded in a high dimensional ambient space. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with degenerate eigenspectrum.

The major part of this thesis revolves around the real algebraic geometry of curves, especially curves of degree six. We use the topological and rigid isotopy classifications of plane sextics to explore the reality of several features associated to each class, such as the bitangents, inflection points, and tensor eigenvectors. We also study the real tensor rank of plane sextics, the construction of quartic surfaces with prescribed topology, and the avoidance locus, which is the locus of all real lines that do not meet a given plane curve. In the case of space sextics, a classical construction relates an important family of these genus 4 curves to the del Pezzo surfaces of degree one. We show that this construction simplifies several problems related to space sextics over the field of real numbers. In particular, we find an example of a space sextic with 120 totally real tritangent planes, which answers a historical problem originating from Arnold Emch in 1928. The last part of this thesis is an algebraic study of a real optimization problem known as Weber problem. We give an explanation and a partial proof for a conjecture on the algebraic degree of the Fermat-Weber point over the field of rational numbers.

Im Rahmen des TERA-Projektes (Turbo Evaluation and Rapid Algorithms) wurde ein neuartiger, hochgradig effizienter probabilistischer Algorithmus zum Lösen polynomialer Gleichungssysteme entwickelt und für den komplexen Fall implementiert. Die Geometrie polarer Varietäten gestattet es, diesen Algorithmus zu einem Verfahren zur Charakterisierung der reellen Lösungsmengen polynomialer Gleichungssysteme zu erweitern. Ziel dieser Arbeit ist es, eine Implementierung dieses Verfahrens zur Bestimmung reeller Lösungen auf eine Klasse von Beispielproblemen anzuwenden. Dabei wurde Wert darauf gelegt, dass diese Beispiele reale, praxisbezogene Anwendungen besitzen. Diese Anforderung ist z.B. für polynomiale Gleichungssysteme erfüllt, die sich aus dem Entwurf von schnellen Wavelet-Transformationen ergeben. Die hier betrachteten Wavelet-Transformationen sollen die praktisch wichtigen Eigenschaften der Orthogonalität und Symmetrie besitzen. Die Konstruktion einer solchen Wavelet-Transformation hängt von endlich vielen reellen Parametern ab. Diese Parameter müssen gewisse polynomiale Gleichungen erfüllen. In der veröffentlichten Literatur zu diesem Thema wurden bisher ausschließlich Beispiele mit endlichen Lösungsmengen behandelt. Zur Berechnung dieser Beispiele war es dabei ausreichend, quadratische Gleichungen in einer oder zwei Variablen zu lösen. Zur Charakterisierung der reellen Lösungsmenge eines polynomialen Gleichungssystems ist es ein erster Schritt, in jeder reellen Zusammenhangskomponente mindestens einen Punkt aufzufinden. Schon dies ist ein intrinsisch schweres Problem. Es stellt sich heraus, dass der Algorithmus des TERA-Projektes zur Lösung dieser Aufgabe bestens geeignet ist und daher eine größere Anzahl von Beispielproblemen lösen kann als die besten kommerziell erhältlichen Lösungsverfahren. / As a result of the TERA-project on Turbo Evaluation and Rapid Algorithms a new type, highly efficient probabilistic algorithm for the solution of systems of polynomial equations was developed and implemented for the complex case. The geometry of polar varieties allows to extend this algorithm to a method for the characterization of the real solution set of systems of polynomial equations. The aim of this work is to apply an implementation of this method for the determination of real solutions to a class of example problems. Special emphasis was placed on the fact that those example problems possess real-life, practical applications. This requirement is satisfied for the systems of polynomial equations that result from the design of fast wavelet transforms. The wavelet transforms considered here shall possess the practical important properties of symmetry and orthogonality. The specification of such a wavelet transform depends on a finite number of real parameters. Those parameters have to obey certain polynomial equations. In the literature published on this topic, only example problems with a finite solution set were presented. For the computation of those examples it was sufficient to solve quadratic equations in one or two variables. To characterize the set of real solutions of a system of polynomial equations it is a first step to find at least one point in each connected component. Already this is an intrinsically hard problem. It turns out that the algorithm of the TERA-project performes very well with this task and is able to solve a larger number of examples than the best known commercial polynomial solvers.

This thesis analyses the semialgebraic sets, that is, a finite union of solu- tions to a finite sequence of polynomial inequalities. We introduce a notion of cylindrical algebraic decomposition as a tool for the construction of a semialge- braic stratification and a triangulation of a semialgebraic set. On this basis, we prove several important and well-known results of real algebraic geometry, such as Hardt's semialgebraic triviality or Sard's theorem. Drawing on Morse theory, we finally give a proof of a Thom-Milnor bound for a sum of Betti numbers of a real algebraic set. 1

Afin d’étudier les classes d'isotopie rigide des courbes rationnelles nodales de degré 5 dans RPP, nous associons à chaque quintique avec un point double réel marque une courbe trigonale dans la surface de Hirzebruch Sigma3 et le dessin reel nodal correspondant dans CP/(z mapsto bar{z}). Les dessins sont des versions réelles, proposées par S. Orevkov dans cite{Orevkov}, des dessins d'enfants de Grothendieck. Un dessin est un graphe contenu dans une surface topologique, muni d'une certaine structure supplémentaire. Dans cette thèse, nous étudions les propriétés combinatoires et les recompositions des dessins correspondants aux courbes trigonales nodales C subset Sigma dans les surfaces réglées réelles Sigma . Les dessins uninodaux sur une surface a bord quelconque et les dessins nodaux sur le disque peuvent être décomposés en blocs correspondant aux dessins cubiques sur le disque D2 , ce qui conduit a une classification des ces dessins. La classification des dessins considérés mène à une classification à isotopie rigide des courbes rationnelles nodales de degré 5 dans RPP. / In order to study the rigid isotopy classes of nodal rational curves of degree $5$ in $\RPP$, we associate to every real rational quintic curve with a marked real nodal point a trigonal curve in the Hirzebruch surface $\Sigma_3$ and the corresponding nodal real dessin on~$\CP/(z\mapsto\bar{z})$. The dessins are real versions, proposed by S. Orevkov~\cite{Orevkov}, of Grothendieck's {\it dessins d'enfants}. The {\it dessins} are graphs embedded in a topological surface and endowed with a certain additional structure. We study the combinatorial properties and decompositions of dessins corresponding to real nodal trigonal curves~$C\subset \Sigma$ in real ruled surfaces~$\Sigma$. Uninodal dessins in any surface with non-empty boundary and nodal dessins in the disk can be decomposed in blocks corresponding to cubic dessins in the disk~$\mathbf{D}^2$, which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of real rational quintics in~$\RPP$.

<p>Studying properties of random polynomials has marked a shift in algebraic geometry. Instead of worst-case analysis, which often leads to overly pessimistic perspectives, randomness helps perform average-case analysis, and thus obtain a more realistic view. Also, via Erdos' astonishing 'probabilistic method', one can potentially obtain deterministic results by introducing randomness into a question that apriori had nothing to do with randomness. </p> <p><br></p> <p>In this thesis, we study topological questions in real algebraic geometry, o-minimal geometry and random algebraic geometry, with motivation from incidence combinatorics. Specifically, we prove results along two different threads:</p> <p><br></p> <p>1. Topology of semi-algebraic and definable (over any o-minimal structure over R) sets, in both deterministic and random settings.</p><p>2. Topology of random hypersurface arrangements. In this case, we also prove a result that could be of independent interest in random graph theory.</p> <p><br></p> <p>Towards the first thread, motivated by applications in o-minimal incidence combinatorics, we prove bounds (both deterministic and random) on the topological complexity (as measured by the Betti numbers) of general definable hypersurfaces restricted to algebraic sets. Given any sequence of hypersurfaces, we show that there exists a definable hypersurface G, and a sequence of polynomials, such that each manifold in the sequence of hypersurfaces appears as a component of G restricted to the zero set of some polynomial in the sequence of polynomials. This shows that the topology of the intersection of a definable hypersurface and an algebraic set can be made <i>arbitrarily pathological</i>. On the other hand, we show that for random polynomials, the Betti numbers of the restriction of the zero set of a random polynomial to any definable set deviates from a Bezout-type bound with <i>bounded probability</i>.</p> <p><br></p> <p>Progress in o-minimal incidence combinatorics has lagged behind the developments in incidence combinatorics in the algebraic case due to the absence of an o-minimal version of the Guth-Katz <i>polynomial partitioning</i> theorem, and the first part of our work explains why this is so difficult. However, our average result shows that if we can prove that the measure of the set of polynomials which satisfy a certain property necessary for polynomial partitioning is suitably bounded from below, by the <i>probabilistic method</i>, we get an o-minimal polynomial partitioning theorem. This would be a tremendous breakthrough and would enable progress on multiple fronts in model theoretic combinatorics. </p> <p><br></p> <p>Along the second thread, we have studied the average Betti numbers of <i>random hypersurface arrangements</i>. Specifically, we study how the average Betti numbers of a finite arrangement of random hypersurfaces grows in terms of the degrees of the polynomials in the arrangement, as well as the number of polynomials. This is proved using a random Mayer-Vietoris spectral sequence argument. We supplement this result with a better bound on the average Betti numbers when one considers an <i>arrangement of quadrics</i>. This question turns out to be equivalent to studying the expected number of connected components of a certain <i>random graph model</i>, which has not been studied before, and thus could be of independent interest. While our motivation once again was incidence combinatorics, we obtained the first bounds on the topology of arrangements of random hypersurfaces, with an unexpected bonus of a result in random graphs.</p>

In 2020 Parusiński and Rond proved that every algebraic set X ⊂ R^n is homeomorphic to an algebraic set X’ ⊂ R^n which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called R|Q-regularity of points of a Q-algebraic set X ⊂ R^n. This definition suggests a natural notion of a Q-nonsingular Q-algebraic set X ⊂ R^n. The study of Q-nonsingular Q-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce Q-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions ‘over Q’ of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 &amp; 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending ‘over Q’ the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of Q-nonsingular Q-algebraic models of Nash manifolds over every real closed field and an answer to the Q-algebrization problem for germs of an isolated algebraic singularity. Appendices A &amp; B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.

Un problème de géométrie algébrique réelle s'exprime sous forme d’un système d’équations et d’inéquations polynomiales, dont l’ensemble des solutions est un ensemble semi-algébrique. L'objectif de cette thèse est de montrer comment les algorithmes de ce domaine peuvent être décrits formellement dans le langage du système de preuve Coq.Un premier résultat est la définition formelle et la certification de l’algorithme de transformation de Newton présentée dans la thèse d'A. Bostan. Ce travail fait intervenir non seulement des polynômes, mais également des séries formelles tronquées. Un deuxième résultat est la description d'un type de donnée représentant les ensembles semi-algébriques. Un ensemble semialgébrique est représenté par une formule logique du premier ordre basée sur des comparaisons entre expressions polynomiales multivariées. Pour ce type de données, nous montrons comment obtenir les différentes opérations ensemblistes et allons jusqu'à décrire les fonctions semi-algébriques. Pour toutes ces étapes, nous fournissons des preuves formelles vérifiées à l'aide de Coq. Enfin, nous montrons également comment la continuité des fonctions semi-algébrique peut être décrite, mais sans en fournir une preuve formelle complète. / A real algebraic geometry problem is expressed as a system of polynomial equations and inequalities, and the set of solutions are semi-algebraic sets. The objective of this thesis is to show how the algorithms of this domain can be formally described in the language of the Coq proof system. A first result is the formal definition and certification of the Newton transformation algorithm presented in A. Bostan's thesis. This work involves not only polynomials, but also truncated formal series. A second result is the description of a data type representing semi-algebraic sets. A semi-algebraic set is represented by a first-order logical formula based on comparisons between multivariate polynomial expressions. For this type of data, we show how to obtain the different set operations all the way to describing semialgebraic functions. For all these steps, we provide formal proofs verified with Coq. Finally, we also show how the continuity of semi-algebraic functions can be described, but without providing a fully formalized proof.

Cette thèse porte sur la commande H∞ par loop-shaping pour les systèmes linéaires à temps invariant d'ordre faible avec ou sans retard et dépendant de paramètres inconnus. L'objectif est d'obtenir des correcteurs H∞ paramétriques, c'est-à-dire dépendant explicitement des paramètres inconnus, pour application à des viseurs gyrostabilisés.L'existence de ces paramètres inconnus ne permet plus l'utilisation des techniques numériques classiques pour la résolution du problème H∞ par loop-shaping. Nous avons alors développé une nouvelle méthodologie permettant de traiter les systèmes linéaires de dimension finie grâce à l'utilissation de techniques modernes de calcul formel dédiées à la résolution des systèmes polynomiaux (bases de Gröbner, variétés discriminantes, etc.).Une telle approche présente de multiples avantages: étude de sensibilités du critère H∞ par rapport aux paramètres, identification de valeurs de paramètres singulières ou remarquables, conception de correcteurs explicites optimaux/robustes, certification numérique des calculs, etc. De plus, nous montrons que cette approche peut s'étendre à une classe de systèmes à retard.Plus généralement, cette thèse s'appuie sur une étude symbolique des équations de Riccati algébriques. Les méthodologies génériques développées ici peuvent s'étendre à de nombreux problèmes de l'automatique, notamment la commande LQG, le filtrage de Kalman ou invariant. / This PhD thesis deals with the H∞ loop-shaping design for low order linear time invariant systems depending on unknown parameters. The objective of the PhD thesis is to obtain parametric H∞ controllers, i.e. controllers which depend explicitly on the unknown model parameters, and to apply them to the stabilization of gyrostabilized sights.Due to the unknown parameters, no numerical algorithm can solve the robust control problem. Using modern symbolic techniques dedicated to the solving of polynomial systems (Gröbner bases, discriminant varieties, etc.), we develop a new methodology to solve this problem for finite-dimensional linear systems.This approach shows several advantages : we can study the sensibilities of the H∞ criterion to the parameter variations, identify singular or remarquable values of the parameters, compute controllers which depend explicitly on the parameters, certify the numerical computations, etc. Furthermore, we show that this approach can be extended to a class of linear time-delay systems.More generally, this PhD thesis develops an algebraic approach for the study of algebraic Riccati equations. Thus, the methodology obtained can be extended to many different problems such as LQG control and Kalman or invariant filtering.