Metric and Random
               Algebraic Geometry
              Paul Breiding and Antonio Lerario
         Author’s addresses:
         Paul Breiding, Universit¨at Osnabruc¨ k, pbreiding@uni-osnabrueck.de.
         Antonio Lerario, SISSA, lerario@sissa.it.
        P. Breiding has been funded by the Deutsche Forschungsgemeinschaft
        (DFG, German Research Foundation) – Projektnummer 445466444.
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                    Contents
                    1 How many zeros of a polynomial are real?                                                1
                        1.1   Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       3
                        1.2   Real discriminants     . . . . . . . . . . . . . . . . . . . . . . . . . . .    7
                        1.3   Reasonable probability distributions . . . . . . . . . . . . . . . . . .        9
                        1.4   The Kostlan distribution . . . . . . . . . . . . . . . . . . . . . . . .      12
                        1.5   Expected properties . . . . . . . . . . . . . . . . . . . . . . . . . . .     13
                    2 Riemannian manifolds and probability                                                  16
                        2.1   Basics from differential geometry      . . . . . . . . . . . . . . . . . . .  16
                              2.1.1   Basic notions and examples . . . . . . . . . . . . . . . . . .        16
                              2.1.2   Vector bundles     . . . . . . . . . . . . . . . . . . . . . . . . .   23
                        2.2   The Riemannian volume . . . . . . . . . . . . . . . . . . . . . . . .         26
                              2.2.1   Riemannian manifolds and integrals . . . . . . . . . . . . . .        26
                              2.2.2   Measure theoretic considerations       . . . . . . . . . . . . . . .  29
                              2.2.3   The coarea formula . . . . . . . . . . . . . . . . . . . . . . .      30
                              2.2.4   Isometries and Riemannian submersions . . . . . . . . . . .           32
                              2.2.5   Volume of the sphere and projective space . . . . . . . . . .         33
                              2.2.6   Volume of the Orthogonal and Unitary group . . . . . . . .            36
                    3 Semialgebraic geometry                                                                38
                        3.1   Semialgebraic sets and functions . . . . . . . . . . . . . . . . . . . .      38
                        3.2   Decomposition of semialgebraic sets and their stratification . . . . .        41
                        3.3   Cohomology of semialgebraic sets . . . . . . . . . . . . . . . . . . .        47
                        3.4   The mapping cyclinder of semialgebraic functions . . . . . . . . . .          51
                        3.5   Semialgebraic triviality . . . . . . . . . . . . . . . . . . . . . . . . .     56
                    4 Topology of algebraic sets                                                            60
                        4.1   Thom’s Isotopy Lemma . . . . . . . . . . . . . . . . . . . . . . . . .         60
                        4.2   Abound on the Betti numbers of real algebraic sets . . . . . . . . .          66
                        4.3   The fundamental class of a real algebraic set . . . . . . . . . . . . .       73
                    5 The Kac-Rice formula                                                                  79
                        5.1   The Kac-Rice formula in Euclidean Space          . . . . . . . . . . . . . .  80
                        5.2   Root density of Kac polynomials        . . . . . . . . . . . . . . . . . . .  84
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                        5.3   The Kac-Rice formula for random maps on manifolds . . . . . . . .              88
                        5.4   Root density of systems of Kostlan polynomials . . . . . . . . . . .           90
                        5.5   Random sections of vector bundles         . . . . . . . . . . . . . . . . . .  93
                        5.6   There are 6√2−3 lines on a real cubic surface . . . . . . . . . . . .          97
                    6 Homogeneous spaces and integral geometry                                              101
                        6.1   Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
                        6.2   The Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
                        6.3   Volumes of homogeneous spaces . . . . . . . . . . . . . . . . . . . . 103
                        6.4   Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
                        6.5   Integral geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
                        6.6   Probabilistic intersection theory in projective space . . . . . . . . . 109
                        6.7   Proof of the integral geometry formula . . . . . . . . . . . . . . . . 112
                    7 Representation theory                                                                 116
                        7.1   Invariant Hermitian structures . . . . . . . . . . . . . . . . . . . . . 121
                        7.2   Classification of real invariant inner products . . . . . . . . . . . . . 123
                    8 Invariant inner products on the space of polynomials                                  130
                        8.1   Complex invariant distributions . . . . . . . . . . . . . . . . . . . . 130
                        8.2   Real invariant distributions     . . . . . . . . . . . . . . . . . . . . . . 132
                              8.2.1   Spherical harmonics      . . . . . . . . . . . . . . . . . . . . . . 134
                    9 Discriminants                                                                         140
                        9.1   The main theorem of elimination theory . . . . . . . . . . . . . . . 140
                        9.2   The discriminant in the space of complex polynomials . . . . . . . . 143
                        9.3   The discriminant in the space of real polynomials . . . . . . . . . . 152
                        9.4   The discriminant in the space of real quadrics . . . . . . . . . . . . 157
                        9.5   The distance to the discriminant . . . . . . . . . . . . . . . . . . . . 161