Algebraic Methods for Dynamical Systems and Optimisation Nidhi Kaihnsa Von der Fakult¨atf¨urMathematik und Informatik der Universit¨atLeipzig angenommene DISSERTATION zur Erlangung des akademischen Grades DOCTOR RERUM NATURALIUM (Dr.rer.nat.) im Fachgebiet Mathematik vorgelegt von Diplommathematikerin Nidhi Kaihnsa geboren am 10.12.1992 in India Die Annahme der Dissertation wurde empfohlen von: Gutachter 1. Professor Dr. Bernd Sturmfels (MPI, MIS, Leipzig) 2. Professor Dr. Timo de Wolff (TU Braunschweig) Die Verleihung des akademischen Grades erfolgt mit Bestehen der Verteidigung am 03.07.2019 mit dem Gesamtpr¨adikat magna cum laude. ii Abstract 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 at- tainable 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 com- pute 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 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 Pn we look at its image X◦r under the map that r r takes each point [x0 : ::: : xn] in X to its coordinate-wise r-th power [x0 : ::: : xn]: 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. Contents List of Figures vi List of Tables viii 1 Introduction 1 1.1 Algebraic Geometry and Chemical Reaction Networks . .1 1.2 Optimisation and Polynomials . .2 1.3 Understanding Powers of Varieties . .4 2 Chemistry and Convexity 5 2.1 The Attainable Region . .5 2.1.1 Chemical Reaction Networks . .5 2.1.2 Spectrahedra . .7 2.1.3 Linear Systems . .8 2.1.4 Weakly Reversible Chemical Reaction Networks . 11 2.1.5 Facial Structure . 13 2.2 Representing Convex Hulls . 16 2.2.1 Limiting Faces . 17 2.2.2 Convex Hulls in Bensolve . 21 2.2.3 Boundary of Convex Hulls . 24 2.3 Forward Closed Convex Sets . 30 2.3.1 Planar Case . 30 2.3.2 General Case . 32 2.4 Applications . 34 3 Optimisation and Real Algebraic Geometry 37 3.1 Sixty Four Curves . 37 3.1.1 Discriminantal Transitions . 40 3.1.2 Construction of Representatives . 43 3.1.3 Identifying the Type . 47 3.1.4 Probability Distribution and Experiments . 48 3.1.5 Avoidance Locus . 55 3.1.6 List of Representatives . 61 3.2 Biology and Optimisation . 64 3.2.1 The Mathematical Framework . 65 3.2.2 The Algebraic Optimisation Problem . 67 3.2.3 Experimental Results . 72 4 Coordinate-wise Powers 75 4.1 Degree Formula . 75 4.1.1 Orthostochastic Matrices . 77 4.1.2 Linear Spaces . 79 iv 4.2 Hypersurfaces . 81 4.2.1 The Defining Equation . 81 4.2.2 Duals and Reciprocals of Power Sum Hypersurfaces . 84 4.2.3 From Hypersurfaces to Arbitrary Varieties? . 87 4.3 Linear Spaces . 89 4.3.1 Point Configurations . 90 4.3.2 Degenerate Eigenvalues and Squaring . 91 4.3.3 Squaring Lines and Planes . 93 4.3.4 Squaring in High Ambient Dimensions . 96 5 References 103 v List of Figures 1 Three vertices are the chemical complexes. The labels κi are the rates of reactions. .6 2 Trajectory and its convex hull . 12 3 Face(2) of a curve in 3-space. 14 4 Face(3) with initial point as one vertex of the 3-face for a curve in 4-space. 14 5 Face(3) of a 5-dimensional convex body . 15 6 A Hausdorff convergent sequence of facets F" of A" need not converge to a face F of C. The face F in the left diagram contains a curve point y 2 C which is not extremal in C. The endpoint F of the curve C on the right is an exposed face of C but it is not uniquely exposed. There is no sequence of facets F" of A" that Hausdorff converges to exposed face F .......... 19 7 A sample of points (left) from a space curve and its convex hull (right) . 24 8 Patches on the convex hull. 26 9 Two 2-patches (left) and three 1-patches (right) in the boundary of a 4- dimensional convex body, obtained as the convex hull of a trigonometric curve of degree six. The picture shows the graph G, with five connected components Gi, that is computed by Algorithm 2.2.18. 27 10 The convex hull of a trigonometric curve of degree 14 in 3-space. The bound- ary of this convex body consists of triangles and of 1-patches in a ruled surface of degree 286. 29 11 The Hamiltonian vector field defined by the Trott curve and two of its tra- jectories. 31 12 A pair of ellipses encloses the Trott curve and bounds the attainable region. 32 13 Convex hull of a trajectory of the Van de Vusse reaction and the partition of its boundary . 35 14 Convex hull of a trajectory of a weakly reversible system that is not forward closed . 36 15 The 56 types of smooth plane sextics form a partially ordered set. The colour code indicates whether the real curve divides its Riemann surface. The red curves are dividing, the blue curves are non-dividing, and the purple curves can be either dividing or non-dividing. 39 16 Type (21)2d transitions into Type (21)2nd by turning an oval inside out. 42 17 A sextic of Type (11)7 is constructed by perturbing the union of three quadrics. 45 18 The discriminant divides the Robinson net (42) into 15 components that real- ise four topological types. The green region represents smooth sextics with 10 non-nested ovals. 45 19 Using local perturbations to create sextics that are dividing or non-dividing 46 _ 20 The Edge quartic C and its dual C ; the avoidance locus AC is coloured. 56 21 A sextic curve C with 8 non-nested ovals; its 68 relevant bitangents represent AC . 59 22 A smooth sextic with 10 non-nested ovals whose avoidance locus is empty . 60 23 The effect of cooperativity on a binding curve . 64 vi 24 A molecule with 4 binding sites. 66 25 Minimal molecules for binding polynomial P1, P2, P3 ............. 68 26 Computing cooperativity for a molecule with 3 sites using SCIP ....... 71 27 Minimal molecules for binding polynomials P1, P5, P6 ............. 73 3 28 The coordinate-wise square of the plane V (x0 + x1 + x2 + x3) ⊂ P ...... 83 29 Circles and their coordinate-wise squares . 84 30 The iterated dual-reciprocal DRDR V (f) ⊂ P3 ................ 87 ◦2 ◦2 ◦2 31 Distinction between V (f1 ; f2 ) and V (f1; f2) ................ 88 32 Dependence of L◦2 on the planar point configuration Z ............ 94 vii List of Tables 1 Census of random trigonometric curves in 3-space . 29 2 Rokhlin{Nikulin classification of smooth sextics in the real projective plane . 38 3 Counts of topological types sampled from the U(3)-invariant distribution . 49 4 Sextics with coefficients in {−1012;:::; 1012g uniformly distributed . 50 5 Symmetric sextics with coefficients in {−1012;:::; 1012g uniformly distributed 50 6 Sextics that are determinants of random symmetric matrices with linear entries 50 7 Sextics that are signed sums of n sixth powers of linear forms . 50 8 Computational results for the number of real solutions for inflection points, eigenvectors, bitangents and real rank among the 64 rigid isotopy classes of smooth sextics in 2 ............................... 52 PR 9 Binding polynomials 1 to 8 and the relation of their degree of cooperativity according to the maximal slope of the Hill plot nmax ( = bigger nmax, DPG = 2,3-diphosphoglycerate) .