数的几何讲义

Chapter Ⅰ Minkowski’s Two TheoremsLecture Ⅰ1．Convex sets2．Convex bodies3．Gauge function of a convex body4．Convex bodies with a centreLecture Ⅱ1．Minkowski’s First Theorem2．Lemma on bounded open sets in IRn3．Proof of Minkowski’s First Theorem4．Minkowski’s theorem for the gauge function5．The minimum of the gauge function for an arbitrary lattice in IRn6．ExamplesLecture Ⅲ1．Evaluation of a volume integral2．Discriminant of an irreducible polynomial3．Successive miruma4．Minkowski’s Second Theorem （Theorem 16）Lecture Ⅳ1．A possible method of proof2．A simple example3．A complicated transformation4．Volume of the transformed body5．Proof of Theorem 16 （Minkowski’s Second Theorem）Chapter Ⅱ Linear InequalitiesLecture Ⅴ1．Vector groups2．Construction of a basis3．Relation between different bases for a lattice4．Sub-lattices5．Congruences relative to a sub-lattice6．The number of sub-lattices with given indexLecture Ⅵ1．Local rank of a vector group2．Decomposition of a general vector group3．Characters of vector groups4．Conditions on characters5．Duality theorem for character groups6．Kronecker’s approximation theoremLecture Ⅶ1．Periods of real functions2．Periods of analytic functions3．Periods of entire functions4．Minkowski’s theorem on linear formsLecture Ⅷ1．Completing a given set of vectors to form a basis for a lattice2．Completing a matrix to a unimodular matrix3．A slight extension of Minkowski’s theorem on linear forms4．A limiting case5．A theorem about parquets6．Parquets formed by parallelepipedsLecture Ⅸ1．Products of linear forms2．Product of two linear forms3．Approximation of irrationals4．Product of three linear forms5．Minimum of positive-definite quadrat，ic formsChapter Ⅲ Theory of ReductionLecture Ⅹ1．The problem of reduction2．Space of all matrices3．Minimizing vectors4．Primitive sets5．Construction of a reduced basis6．The First Finiteness Theorem7．Criteria for reduction8．Use of a quadratic gauge function9．Reduction of positive-definite quadratic formsLecture Ⅺ1．Space of symmetric matrices2．Reduction of positive-definite quadratic forms3．Consequences of the reduction conditions4．The case n=25．Reduction of lattices of rank two6．The case n=3Lecture Ⅻ1．Extrema of positive-definite quadratic forms2．Closest packinng of （solid） spheres3．Closest packing in two， three， or four dimensions4．Blichfeldt’s methodLecture ⅩⅢ1．The Second Finiteness Theorem2．An inequality for positive-definite symmetric matrices3．The space PK4．Images of RLecture ⅩⅣ1．Boundary points2．Non overlapping of images3．Space defined by a finite number of conditions4．The Second Finiteness Theorem5．Fundamental region of the space of all matricesLecture ⅩⅤ1．Volume of a fundamental region2．Outline of the proof3．Change of variable4．A new fundamental region5．Integrals over fundamental regions are equal6．Evaluation of the integral7．Generalizations of Minkowski’s First Theorem8．A lower bound for the packing of spheresReferences