Gruber P. Convex and Discrete Geometry

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19 Lattice Polytopes 333 Irreducibility of polynomials has a long history and there is a large body of pertinent results. See, e.g. Schmidt [895], Lidl and Niederreiter [656], Mignotte and Stefanescu [723] and Schinzel [887]. It plays a crucial role in finite geometry, see Hirschfeld [506], combinatorics, see Szönyi [983], and coding, compare Stichtenoth [969]. Surprisingly, Newton polytopes provide an effective tool to describe large classes of (absolute) irreducible polynomials, see Gao [355] and the references there. A remarkable result of Bernstein [100] says that the number of roots of a generic system of polynomial equations over C of the form p1(z1,...,zd) = 0 p2(z1,...,zd) = 0 ................. pd(z1,...,zd) = 0 equals d ! V (N p1 ,...,N pd ). The case d = 2 was anticipated by Minding [731] in 1841 using different terminology, as pointed out by Khovanskiĭ [583]. The case where the Newton polytopes of p1,...,pd all coincide was treated slightly before Bernstein by Kouchnirenko [612]. This result gave rise to several alternative proofs, generalizations and expositions, see Huber and Sturmfels [525, 526], Sturmfels [974], Rojas [853, 854] and the nice book by Sturmfels [975]. In the latter there are described applications to economics (Nash equilibria of n-person games), statistics (random walks on Zd , numerical algorithms for maximum likelihood equations), linear partial differential equations and other areas. This section contains a simple result of Ostrowski [781] which yields an irreducibility criterion, see Gao [355]. Special cases include the Stepanov–Schmidt [895] criterion for irreducibility. Then the result of Minding–Bernstein is given without proof. For more detailed information, see the articles, surveys and books cited earlier. An Irreducibility Criterion The first result is a simple observation of Ostrowski [781] on Newton polytopes, see also Gao [355]. Proposition 19.4. Let p, q : E d → R be real polynomials. Then N pq = N p + Nq. Proof. Since each monomial of the polynomial pq is a sum of products of a monomial of p and a monomial of q, wehave N pq ⊆ N p + Nq. To see the reverse inclusion, it is sufficient to show that each vertex u of the convex lattice polytope N p + Nq is contained in N pq. By Lemma 6.1, u is the sum of a support set of N p and a support set of Nq. Since u is a singleton, these support sets
334 Convex Polytopes have to be singletons too and thus are vertices of N p and Nq, respectively, say v,w. Then u = v +w and this is the only possible representation of u as the sum of a point of N p and a point of Nq. The product of the corresponding monomials bx v1 1 ···xvd d of p and cx w1 1 ···x wd d of q is thus a monomial of the form axu1 1 ···xud d and no other product of a monomial of p and a monomial of q has this form. Hence ax u1 1 ···xud d is the monomial in pq corresponding to u,oru ∈ N pq. Thus N p + Nq ⊆ N pq, concluding the proof. ⊓⊔ A real polynomial on E d is irreducible if it cannot be represented as a product of two real polynomials on E d , each consisting of more than one monomial. Call a convex lattice polytope in E d integer irreducible if it cannot be represented as the sum of two convex lattice polytopes, each consisting of more than one point. As an immediate consequence of the earlier Proposition 19.4, we obtain the following irreducibility criterion, see Gao [355]. Theorem 19.7. Let p : E d → R be a real polynomial with Newton polytope N p. If N p is integer irreducible, then p is irreducible. Remark. Actually, this result holds for any polynomial p in d variables over any field F. In addition, irreducibility of p may be sharpened to absolute irreducibility, i.e. irreducibility in the algebraic closure of F. When N p is not integer irreducible, then p may still be irreducible. An example is provided by the irreducible real polynomial p(x, y) = 1 + y + xy + x 2 + y 2 , whose Newton polytope N p = conv � o,(0, 2), (2, 0) � is integer reducible: conv{o,(0, 2), (2, 0)} =conv{o,(0, 1), (1, 0)}+conv{o,(0, 1), (1, 0)}. Gao and Lauder [356] study the integer irreducibility of convex lattice polytopes. It turns out that the problem to decide whether a convex lattice polytope is integer irreducible is NP-complete. The Stepanov–Schmidt Irreducibility Criterion Clearly, a convex lattice polygon in E 2 , which has an edge of the form [(0, m), (n, 0)] with m and n relatively prime and which is contained in the triangle conv{o,(0, m), (n, 0)}, is integer irreducible. This leads to the following criterion of Stepanov and Schmidt, see W. Schmidt [895], p. 92, and Gao [355]. Corollary 19.2. Let p be a real polynomial in two variables, such that its Newton polygon N p contains an edge of the form [(0, m), (n, 0)], where m and n are relatively prime, and is contained in the triangle conv � o,(0, m), (n, 0) � . Then p is irreducible. For example, the polynomials x 2 + y 3 and x 3 + y 7 + xy+ xy 2 + x 2 y 2 are irreducible.
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Grundlehren der mathematischen Wiss
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Peter M. Gruber Institute of Discre
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VI Preface theory of finite groups
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Contents Preface ..................
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Contents XI 12 Special Convex Bodie
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Contents XIII 29 Packing of Balls a
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2 Convex Functions 1 Convex Functio
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4 Convex Functions x epi f f C y Fi
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6 Convex Functions Together these i
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8 Convex Functions Theorem 1.4. Let
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10 Convex Functions z = f (x) + f
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12 Convex Functions 1.3 Convexity C
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14 Convex Functions Proof (by affin
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16 Convex Functions Minkowski’s I
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18 Convex Functions (ii) Let x > 0.
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20 Convex Functions 2 Convex Functi
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22 Convex Functions | f (z) − f (
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24 Convex Functions First-Order Dif
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26 Convex Functions Remark. More pr
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28 Convex Functions even more is tr
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30 Convex Functions G(x). The proof
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32 Convex Functions by (15), unifor
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34 Convex Functions 2.4 A Stone-Wei
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36 Convex Functions x x Fig. 2.1. S
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38 Convex Functions F is convex. Us
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40 Convex Bodies results, character
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42 Convex Bodies Convex Hulls Given
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44 Convex Bodies The next result is
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46 Convex Bodies C ∩ L ⊥ clearl
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48 Convex Bodies Suppose not. Then,
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50 Convex Bodies The Centrepoint of
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52 Convex Bodies Since G is open, w
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54 Convex Bodies in one of the two
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56 Convex Bodies or geometric infor
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58 Convex Bodies v HC (u, h(u)) (v,
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60 Convex Bodies C D Fig. 4.3. Sepa
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62 Convex Bodies For the proof of t
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64 Convex Bodies in contradiction t
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66 Convex Bodies (6) R(t) ⊆ E d i
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68 Convex Bodies 5 The Boundary of
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70 Convex Bodies for all y ∈ C an
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72 Convex Bodies the existence of s
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74 Convex Bodies where r(·) is a s
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76 Convex Bodies Hence x ∈ conv e
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78 Convex Bodies N = � � X, O Y
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80 Convex Bodies Further topics of
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82 Convex Bodies We are now ready t
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84 Convex Bodies similarly, D j is
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86 Convex Bodies For the proof of t
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88 Convex Bodies The definition of
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90 Convex Bodies If ei ∈ Pi is no
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92 Convex Bodies Let v(·) denote (
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94 Convex Bodies The present sectio
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96 Convex Bodies Proposition 6.5. L
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98 Convex Bodies Thus Second, Minko
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100 Convex Bodies The Valuation Pro
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102 Convex Bodies Minkowski’s the
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104 Convex Bodies To remedy the sit
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106 Convex Bodies Proof. Applying S
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108 Convex Bodies Third, let C be a
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110 Convex Bodies A Problem of Sant
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112 Convex Bodies sets of S. Ifφ c
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114 Convex Bodies = � (−1) |J|
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116 Convex Bodies If P has dimensio
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118 Convex Bodies Since, by Volland
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120 Convex Bodies that is, V is sim
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122 Convex Bodies Let C, D ∈ C su
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124 Convex Bodies P Q . . . . Fig.
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126 Convex Bodies 7.3 Characterizat
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128 Convex Bodies and the Bi have p
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130 Convex Bodies Since the Riemann
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132 Convex Bodies Seventh, o v3 v2
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134 Convex Bodies Mean Width and th
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136 Convex Bodies � χ(C ∩ mD)
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138 Convex Bodies In the following,
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140 Convex Bodies 7.5 Hadwiger’s
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142 Convex Bodies 8.1 The Classical
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144 Convex Bodies C ∩ H(tC ) C D
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146 Convex Bodies Theorem 8.4. Let
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148 Convex Bodies In the following
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150 Convex Bodies Integration impli
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152 Convex Bodies As a consequence
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154 Convex Bodies Proof. � 1 2 w(
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156 Convex Bodies Let ϱ>0 be the m
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158 Convex Bodies Wulff’s Theorem
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160 Convex Bodies Third, by (8) and
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162 Convex Bodies Since the measure
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164 Convex Bodies An application of
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166 Convex Bodies Since f has Lipsc
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168 Convex Bodies Clearly, the inte
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170 Convex Bodies Note that �st :
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172 Convex Bodies Proposition 9.2.
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174 Convex Bodies Proof. Let ε>0.
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176 Convex Bodies The Brunn-Minkows
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178 Convex Bodies 9.3 Schwarz Symme
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180 Convex Bodies Torsional Rigidit
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182 Convex Bodies The proof of (3)
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184 Convex Bodies � �� 2 2
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188 Convex Bodies Both problems inf
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190 Convex Bodies The Area Measure
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192 Convex Bodies halfspace {z : un
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194 Convex Bodies Let σm be the di
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196 Convex Bodies Thus Minkowski’
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198 Convex Bodies Suppose now that
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200 Convex Bodies vector of St at x
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202 Convex Bodies Outer Billiards F
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204 Convex Bodies For more informat
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206 Convex Bodies with P. Thus, it
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208 Convex Bodies Note that ui ∈
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212 Convex Bodies the maximum of th
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214 Convex Bodies (20) mni ≥ 1 λ
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216 Convex Bodies Prove analogous r
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218 Convex Bodies Heuristic Princip
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220 Convex Bodies λC + x νC + z
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222 Convex Bodies To show that (K +
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224 Convex Bodies The integral here
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226 Convex Bodies In the following
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228 Convex Bodies The first such re
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230 Convex Bodies v o Fig. 12.3. Sy
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232 Convex Bodies What Does the Bou
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234 Convex Bodies Corollary 13.1. L
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236 Convex Bodies Hence ν = 0 and
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238 Convex Bodies geometry, see Sei
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240 Convex Bodies We distinguish th
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242 Convex Bodies by Proposition 13
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244 Convex Polytopes The material i
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246 Convex Polytopes V-Polytopes an
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248 Convex Polytopes Convex cones w
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250 Convex Polytopes Generalized Co
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252 Convex Polytopes d 2 inequaliti
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254 Convex Polytopes The Face Latti
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256 Convex Polytopes Then (6) impli
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258 Convex Polytopes P in F. For an
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260 Convex Polytopes Euler’s Poly
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262 Convex Polytopes bijection betw
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264 Convex Polytopes The Converse o
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266 Convex Polytopes (geometric) fa
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268 Convex Polytopes consists preci
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270 Convex Polytopes The Lower and
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272 Convex Polytopes A problem of S
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274 Convex Polytopes The first cond
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276 Convex Polytopes The existence
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278 Convex Polytopes For d = 3 it f
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280 Convex Polytopes 16 Volume of P
Page 293 and 294: 282 Convex Polytopes by (2). Hence
Page 295 and 296: 284 Convex Polytopes (12) The volum
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Page 301 and 302: 290 Convex Polytopes P clearly can
Page 303 and 304: 292 Convex Polytopes 17 Rigidity Ri
Page 305 and 306: 294 Convex Polytopes Fig. 17.2. Cau
Page 307 and 308: 296 Convex Polytopes Call the index
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Page 315 and 316: 304 Convex Polytopes For other pert
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Page 319 and 320: 308 Convex Polytopes 18.3 The Isope
Page 321 and 322: 310 Convex Polytopes 19 Lattice Pol
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Page 329 and 330: 318 Convex Polytopes If the triangl
Page 331 and 332: 320 Convex Polytopes � qP(t) = No
Page 333 and 334: 322 Convex Polytopes Now, take into
Page 335 and 336: 324 Convex Polytopes L � P + (n +
Page 337 and 338: 326 Convex Polytopes V (S) = V .Let
Page 339 and 340: 328 Convex Polytopes (7) ψ|H d = 0
Page 341 and 342: 330 Convex Polytopes Theorem 19.6.
Page 343: 332 Convex Polytopes (15) L(nP) = L
Page 347 and 348: 336 Convex Polytopes economics, von
Page 349 and 350: 338 Convex Polytopes Existence of S
Page 351 and 352: 340 Convex Polytopes Description of
Page 353 and 354: 342 Convex Polytopes (6) u piai xq
Page 355 and 356: 344 Convex Polytopes For x ∈ Bd
Page 357 and 358: 346 Convex Polytopes Rational Polyh
Page 359 and 360: 348 Convex Polytopes Totally Dual I
Page 361 and 362: 350 Convex Polytopes of the vectors
Page 363 and 364: Geometry of Numbers and Aspects of
Page 365 and 366: Extension to Crystallographic Group
Page 367 and 368: Relations Between Different Bases 2
Page 369 and 370: 21 Lattices 359 triangle conv{o,(u,
Page 371 and 372: By definition of bi+1, Thus (6) yie
Page 373 and 374: 21 Lattices 363 A similar, slightly
Page 375 and 376: 21.4 Polar Lattices 21 Lattices 365
Page 377 and 378: The First Fundamental Theorem 22 Mi
Page 379 and 380: 22 Minkowski’s First Fundamental
Page 381 and 382: If 22 Minkowski’s First Fundament
Page 383 and 384: 22 Minkowski’s First Fundamental
Page 385 and 386: Proof. Consider the linear forms l1
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Page 389 and 390: Thus V (C) ≥ V (O) = 2d � �
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24 The Minkowski-Hlawka Theorem 24
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(6) (V (J) ≈) � n�= 0 (7) J
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24 The Minkowski-Hlawka Theorem 389
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The Variance Theorem of Rogers and
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The Selection Theorem of Mahler [68
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26 The Torus Group E d /L 395 where
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Proposition 26.2. E d /L is a compa
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26 The Torus Group E d /L 399 theor
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26 The Torus Group E d /L 401 The m
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The formula to calculate the Jordan
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27 Special Problems in the Geometry
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Asymptotic Estimates 27 Special Pro
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27 Special Problems in the Geometry
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28 Basis Reduction and Polynomial A
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(ii) �b1� ≤2 1 2 (d−1) min
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28 Basis Reduction and Polynomial A
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and so 28 Basis Reduction and Polyn
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Shortest Lattice Vector Problem 28
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and therefore �x − m� 2 = �
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29 Packing of Balls and Positive Qu
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29.3 Error Correcting Codes and Bal
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30 Packing of Convex Bodies 439 The
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30 Packing of Convex Bodies 441 Nex
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Packing and Central Symmetrization
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Existence of Densest Lattice Packin
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30 Packing of Convex Bodies 447 Rem
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A Lower Estimate for the Maximum La
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Lattice Packing and Packing of Tran
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a f b o e 2D c Fig. 30.3. Disc and
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31 Covering with Convex Bodies 455
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32 Tiling with Convex Polytopes 463
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Define 32 Tiling with Convex Polyto
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33.1 Fejes Tóth’s Inequality for
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Noting that 2h 2 v − hhvv = 2π 2
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33 Optimum Quantization 485 The con
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(7) � (x, y) : x ∈ K, 0 ≤ y
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Next, we prove that (18) lim inf n
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Generalizations 33 Optimum Quantiza
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33 Optimum Quantization 493 set wit
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33 Optimum Quantization 495 In (4)
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33 Optimum Quantization 497 encoder
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34 Koebe’s Representation Theorem
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G 34 Koebe’s Representation Theor
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Di+1 34 Koebe’s Representation Th
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References 1. Achatz, H., Kleinschm
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References 515 36. Arias de Reyna,
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References 517 88. Beer, G., Topolo
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References 519 136. Bohr, H., Molle
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References 521 190. Carathéodory,
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References 523 237. Dantzig, G.B.,
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References 525 286. Edmonds, A.L.,
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References 527 337. Florian, A., Ex
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References 529 385. Goodman, J.E.,
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References 531 435. Gruber, P.M., C
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References 533 487. Heil, E., Über
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References 535 534. Hyuga, T., An e
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References 537 582. Khovanskiĭ (Ho
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References 539 632. Lazutkin, V.F.,
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References 541 684. Mani-Levitska,
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References 543 736. Minkowski, H.,
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References 545 783. Pach, J., Agarw
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References 547 835. Rickert, N.W.,
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References 549 891. Schmidt, E., Di
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References 551 941. Skubenko, B.F.,
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References 553 992. Teissier, B., V
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References 555 1042. Zassenhaus, H.
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558 INDEX Bravais classification of
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560 INDEX existence of elementary v
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562 INDEX Kneser-Macbeath theorem,
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564 INDEX rational lattice, 414 rat
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566 INDEX transference theorem of K
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568 AUTHOR INDEX Björner, 265 Blas
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570 AUTHOR INDEX Giles, J., 1 Giles
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572 AUTHOR INDEX Kuczma, 13, 17 Kü
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574 AUTHOR INDEX Pick, 316 Pisier,
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576 AUTHOR INDEX Tanemura, 464 Tarj
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578 List of Symbols sch, schL Schwa
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289. Manin: Gauge Field Theory and
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