Polynomial Optimization and Discrete Geometry Advisor : Christine Bachoc Rujuta .S. Joshi 01/07/2015 1 Acknowledgements I would like to thank Professor Christine Bachoc, my Advisor for providing guidence and all the support. I am also grateful to her for encouraging me to pursue this thesis topic .I would also like to thank my family for their constant support. Contents 1 Introduction3 2 Preliminaries4 2.0.1 Measure theory................................4 2.0.2 Algebraic Geometry.............................5 2.0.3 Linear Functionals..............................6 3 Polynomial Optimization8 3.1 Semidefinite Optimization..............................8 3.1.1 Semidefinite Program.............................8 3.1.2 Applications of Semidefinite Programs to Combinatorial Problems....9 3.1.3 Lovász sandwich inequalities......................... 10 3.2 Sum of squares.................................... 13 3.2.1 Relation between sum of squares and being positive............ 13 3.2.2 Lasserre Hierarchy.............................. 15 3.3 Moments........................................ 17 4 Kissing Number 21 4.1 Spherical Harmonics................................. 21 4.2 Gegenbauer Polynomials............................... 25 4.3 Kissing Number.................................... 26 5 Triangle Packing 29 5.1 Appendix....................................... 36 2 Chapter 1 Introduction Problem Statement :How many non overlapping regular tetrahedra, having a common ver- tex can be arranged in R3? Solution to this problem (say T(3)) is known to satisfy 20 ≤ T (3) ≤ 22). Thesis aims at trying to solve the above problem with the help of polynomial optimization. Thesis is divided into 4 parts. The first part explains basic concepts required further. Second part deals with the theory of polynomial optimization. In part 3 we study spherical harmonics and apply polynomial optimization to the kissing number problem.Eventually in part 4 we try to bound T(3) with tools developed so far. 3 Chapter 2 Preliminaries Some concepts have been introduced ,which will be used later in the thesis . Definition 2.0.0.1. A polynomial optimization problem is to optimize value of f 2 R[x1; ::::; xn] over a set K described by some g1; :::; gm 2 R[x1; ; ; ; xn] .Let us consider computing infimum n of a polynomial f 2 R[x1; :::; xn] over K = fx 2 R j g1(x) ≥ 0; :::::; gm(x) ≥ 0g. fmin := infff(x) j g1(x) ≥ 0; :::; gm(x) ≥ 0g (2.1) n n Pn Notations : Let N = N [ f0g; t ≥ 0; Nt := f(α1; ::; αn) 2 N j i=1 αi ≤ tg; R[x] := n P α α1 αn R[x1; :::; xn]. Let α 2 N (say α = (α1; ::; αn)) ; jαj := αi. Then x = x1 :::xn ; α R[x]t := ff 2 R[x] j degf ≤ tg ; [x]t := (x )jα|≤t (In some fixed order) Now if f 2 R[x]t: Then deg f ≤ t. Therefore coefficient of f can be expressed in a vector form [f] = [f ] n : α α2Nt α t fα coefficient of x (in same order as above) So f = [f] [x]t. Let I be an ideal in R[x] ,then It = ff 2 I j degf ≤ tg 2.0.1 Measure theory Let X be a set. If 2X is collection of subsets of X. Definition 2.0.1.1. σ - algebra Σ : Let Σ ⊆ 2X then Σ is a σ − algebra if 1. X 2 Σ 2. if A 2 Σ; then Xn A in Σ 3. Σ is closed under countable union. i.e If A1;A2; ::: 2 Σ then [i2NAi 2 Σ Definition 2.0.1.2. Measure µ : Measure µ on a set X (with σ − algebra Σ) is a function from Σ to R [ f1g satisfying 1. µ(A) ≥ 0 8 A 2 Σ 2. µ(φ) = 0 3. Let A ;A ; ::: be countable pairwise disjoint subsets of X in Σ, then µ([ A ) = P µ(A ) 1 2 i2N i i2N i 4 CHAPTER 2. PRELIMINARIES 5 Definition 2.0.1.3. Borel measure on a set X : X is locally compact and Hausdorff space.Let Σ be the smallest σ − algebra containing all open sets in X.And a measure µ defined on this σ − algebra is a borel measure. Definition 2.0.1.4. Dirac measure :(let X be set with given σ − algebra Σ).Then dirac measure w.r.t. a fixed point x 2 X is µ(A) = 0 if x2 = A = 1 if x 2 A where A 2 Σ. Definition 2.0.1.5. Probability measure : µ measure on X with σ − algebra Σ is a prob- ability measure if µ takes values in [0; 1] and µ(X) = 1. 2.0.2 Algebraic Geometry Let I be an ideal in R[x] . p m Definition 2.0.2.1.p Radical of I : I :=ff 2 R[x] j f 2 I for some m ≥ 1g. I is said to a radical ideal if I = I. p R 2m 2 2 Definition 2.0.2.2. Real radical of I : I := ff 2 R[x] j f + pp1 + ::: + pk 2 I for some R m ≥ 1 and p1; ::; pk 2 R[x]g. I is said to be a real radical ideal if I = I. n Definition 2.0.2.3. VC(I) : f(a1; :::; an) 2 C j f(a1; :::; an) = 0 8 f 2 Ig. It is called a complex variety. n n Definition 2.0.2.4. VR(I) : f(a1; :::; an) 2 R j f(a1; :::; an) = 0 8 f 2 Ig = VC(I) \ R .It is a called a real variety. Definition 2.0.2.5. I(VC(I)):= ff 2 R[x] j f(a1; :::; an) = 0 8 (a1; :::; an) 2 VC(I)g. Definition 2.0.2.6. I(VR(I)):= ff 2 R[x] j f(a1; :::; an) = 0 8 (a1; :::; an) 2 VR(I)g. p Lemma 2.0.2.7. I ⊆ I ⊆ I(VC(I)) p p Proof. f 2 I ) f 1 2 I;Therefore f 2 I ) f 2 I: f 2 I ) f m 2 I for some m ≥ 1: m Therefore f (a1; :::; an) = 0 8 (a1; :::; an) 2 VC(I) ) f(a1; :::; an) = 0 8 (a1; :::; an) 2 VC(I) ) f 2 I(VC(I)) p R Lemma 2.0.2.8. I ⊆ I ⊆ I(VR(I)) p p 2 R R 2m 2 2 Proof. f 2 I ) f 2 I; Therefore f 2 I ) f 2 I: f 2 I ) f + p1 + ::: + pk 2 I . Let 2m 2 2 n (a1; :::; an) 2 VR(I) then (f +p1+:::+pk)(a1; :::; an) = 0 but (a1; :::; an) 2 R ) pi(a1; :::; an) 2 m 2m R 8 i and f (a1; :::; an) 2 R ) f (a1; :::; an) = 0 ) f(a1; :::; an) = 0 ) f 2 I(VR(I)) Theorem 2.0.2.9.p Hilbert’s Nulltellensatzp and Real Nullstellensatz thm: I ideal in R R[x]: Then I = I(VC(I)) and I = I(VR(I)).(For Hilbert’s Nulltellensatz refer to Serre Lang’s and for real Nulltellensatz refer [3] ) Lemma 2.0.2.10. I ⊆ I(VC(I)) ⊆ I(VR(I)). T n Proof. VR(I): VC(I) R Therefore f 2 I(VC(I)) ) f(a1; :::; an) = 0 8 (a1; :::; an) 2 VC(I) which implies f(a1; :::; an) = 08(a1; :::; an) 2 VR(I) ) f 2 I(VR(I)). Therefore, I ⊆ I(VC(I)) ⊆ I(VR(I)). CHAPTER 2. PRELIMINARIES 6 Theorem 2.0.2.11. If I is a real radical ideal and j VR(I) j< 1, then VC(I) = VR(I). p p R Proof. If I is a real radical ideal then I ⊆ I(VC(I)) ⊆ I = I. It implies I(VC(I)) = I = I = I(VR(I)) =) I is radical and I(VC(I)) = I(VR(I)). Now if I is a real radical and jVR(I)j < 1,then VR(I) = VC(J) for some ideal J.(* if jVR(I)j = 1 say a = (a1; :::; an) = VR(I),then xi −ai 2 R[x1; :::; xn] 8 1 ≤ i ≤ n and VC((x1 −a1; ::::; xn −an)) = VR(I). Now if jVR(I)j = m < 1 then we get ideals J1; :::; Jm such that each point in VR(I) = VC(Ji) for some 1 ≤ j ≤ m and therefore VR(I) = VC(J1:::Jm)) and so I(VC(I)) = I(VR(I)) = I(VC(J)). VC(I(VC(J))) = VC(J) (* VC(J) ⊆ VC(I(VC(J))) and if ap2 VC(I(VpC(J))) then 8 f 2 I(VC(J)); f(a) = 0. But by Hilbert Nullstellensatz I(VC(J)) = J.J ⊆ J;Therefore 8 f 2 J; f(a) = 0. So a 2 VC(J)). Similarly VC(I(VC(I))) = VC(I). So VC(I) = VC(J) = VR(I). Proposition 2.0.2.12. Let I be an ideal in R[x1; :::; xn].Then j VC(I) j< 1 iff R[x1; :::; xn]=I is finite dimensional as a vector space.(For proof refer to [1]) Interpolation Polynomials : n Theorem 2.0.2.13. Let V ⊆ R be finite set.Then there exist polynomials pv 2 R[x1; :::; xn] 8 v 2 V satisfying pv(u) = δu;v 8 u; v 2 V .Then we have that for any polynomial f 2 R[x1; ::; xn] X f − f(v)pv 2 I(VC(I)) (2.2) v 2 VC(I) Proof. Fix v.8u 6= v 9 component iu such that v(iu) 6= u(iu). Define Y pv := (x(iu) − u(iu))=(p(iu) − u(iu)) (2.3) u 2 VC(I)nv According to this definition pv(v) = 1 and pv(u) = 0 8 u 6= v 2 V (I).Let f 2 R[x1; ::; xn].For P P C any u in VC(I) we have (f − v 2 V (I) f(v)pv)(u) = f(u)− v 2 V (I) f(v)pv(u) = f(u)−f(u) = CP C 0.So by definition of I(V (I)); f − f(v)pv 2 I(V (I)).