Number theory, geometry and algebra - Dynamics-approx.jku.at

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Proof. Let P be a minimal polynomial over over Z, so that P(ξ) = 0 (P is then irreducible over Q). For p,q ∈ Z and q > 0 we have ( p ( P(ξ)−P = ξ − q) p ) P ′ (ξ 0 ) q ] [ ] [ ) ) where ξ 0 ∈ ξ, p (resp. p ,ξ ). Then P( p ≠ 0 and q P( n p ∈ Z. We thus q q q q ( ) ∣ have the estimate P p ∣∣ ≥ 1 ∣ . We now choose c so that q q ∣P ′ (ξ n 0 ) ∣ < 1 if c ∣ |ξ 0 −ξ| ≤ 1. Then ∣ξ − p q∣ > c as claimed. q n 2 Geometry 2.1 Triangles: We begin with one of the simplest, but richest of geometrical figures—the triangle. A triangle is determined by its three vertices A, B and C. (Figure 1). It is then denoted by ABC. Normally we shall assume that it is nondegenerate i.e. that A, B and C are not collinear. This can be expressed analytically as the statement that the vectors x B − x B and x C − x A are linearly independent i.e. that there are no non-trivial pairs (λ,µ) of scalars so that λ(x B −x A )+µ(x C −x A ) = 0. (non-trivial means that either λ ≠ 0 or µ ≠ 0). We can rewrite this equation in the form λ 1 x A +λ 2 x B +λ 3 x C = 0 where λ 1 = −(λ+µ), λ 2 = λ,λ 3 = µ. Thisleadstothefollowingmoresymmetricdescriptionofthenon-degeneracy of ABC: the triangle is non-degenerate if and only if there is no triple (λ 1 ,λ 2 ,λ 3 ) of scalars so that λ 1 +λ 2 +λ 3 = 0 and λ 1 x A +λ 2 x B +λ 3 x C = 0. The vectors x A , x B and x C are then said to be affinely independent. In this case, any point x in R 2 can be written as λ 1 x A +λ 2 x B +λ 3 x C 50
where λ 1 + λ 2 + λ 3 = 1. Before proving this we first note that these λ are uniquely determined by x. for if we have a second such representation, say with coefficients µ 1 , µ 2 , µ 3 , then substracting leads to the equation (λ 1 −µ 1 )x A +(λ 2 −µ 2 )x B +(λ 3 −µ 3 )x C = 0 where the sum of the coefficients is zero and the λ’s nand the µ’s coincide. We are therefore justified in calling the λ’s the barycentric coordinates of x with respect to A, B and C. We now turn to the existence: suppose that x is the coordinate vector of the point P and produce AP to meet BC at the point Q (see figure 2). We are tacitly assuming that P is distinct from A and that AP is not parallel to BC - these two special cases are simpler. There are scalars λ, µ so that x Q = λx B +(1−λ)x C and Then we have x P = µx A +(1−µ)x Q . x P = µx A +(1−µ)λx B +(1−µ)(1−λ)x C and the sum of the coefficients is µ+(1−µ)[λ+(1−λ)] = 1. From this we see that the barycentric coordinates have the following geometrical interpretation. For simplicity we shall assume that the point P lies within the triangle (i.e. that its barycentric coordinates are all positive). If we refer to figure 2, we see that the coefficient λ 1 of x A is the ratio |PQ| of |AQ| the lengths of PQ and AQ and we see from figure 3 that this is the ratio of the distance of P resp. of A to the line BC. The point x with barycentric coordinates (λ 1 ,λ 2 ,λ 3 ) can be regarded as the centroid of a physical system consisting f three particles A, B and C with masses λ 1 , λ 2 and λ 3 respectively. The important case of equal masses produces the centroid i.e. the point M with x M + 1(x 3 A+x B +x C ). We see from the above that, as expected, M lies two thirds of the way along the line from A to the midpoint of BC (figure 4) (for x M = 1x 3 A + 3 2 (x B+x C ). 2 Theexistence ofbarycentric coordinatescanbederived inamoreanalytic fashion as follows: since (x B −x A ) and (x C −x A ) are linearly independent, they span R 2 . Hence there exist λ, µ so that x P −x A = λ(x B −x A )+µ(x C −x A ) 51
Page 1 and 2: Number theory, geometry and algebra
Page 3 and 4: Applications: 1. b-adic development
Page 5 and 6: Proof. 4 is non-prime and has the p
Page 7 and 8: We show that α i = β i for each i
Page 9 and 10: We round up this chapter with some
Page 11 and 12: Definition: Ein Integritätsbereich
Page 13 and 14: a = b (modm), c = d (modm) ⇒ a+c
Page 15 and 16: Example The equation 2x = 3 (mod4)
Page 17 and 18: for each i. x is then a solution of
Page 19 and 20: 1.5 Arithmetical functions: In this
Page 21 and 22: σ k (n) = ∏ i τ(n) = ∏ i p k(
Page 23 and 24: is perfect. For σ(m) = σ[(2 p−1
Page 25 and 26: Remark Such an element is called a
Page 27 and 28: Higher order equations: Now let m b
Page 29 and 30: Example We calculate ( −3 ) (p
Page 31 and 32: The Legendre symbol can be generali
Page 33 and 34: [ 2 0 a 0 − d 2 ] if d = 0 (mod 4
Page 35 and 36: Proof. ⇐: Choose a solution x and
Page 37 and 38: Proposition 27 A natural number n i
Page 39 and 40: p k = a k p k−1 +p k−2 (97) q k
Page 41 and 42: The convergence follows from the fa
Page 43 and 44: Proof. Let p = [a q 0;a 1 ,...,a n
Page 45 and 46: Proof. Proof of the main result ⇒
Page 47 and 48: The Farey sequence F \ is the finit
Page 49: . If this is not the case, then 1 1
Page 53 and 54: Now x P −x R = −(1−t 3 )x A +
Page 55 and 56: 1 This it suffices to substitute b+
Page 57 and 58: case is excluded) and convex (i.e.
Page 59 and 60: Cyclic quadrilaterals In general, a
Page 61 and 62: The above result suggests the follo
Page 63 and 64: from which the result can be read o
Page 65 and 66: where the Z k ’s are the eigenvec
Page 67 and 68: 2.5 Circles The circle with centre
Page 69 and 70: which simplifies to (λ 1 −λ 2 )
Page 71 and 72: and [ ξ1 ξ 2 ] . The condition im
Page 73 and 74: • rotation plus reflection = glid
Page 75 and 76: If C 1 and C 2 are circles with dis
Page 77 and 78: Inordertodothis, wetranslateC 1 alo
Page 79 and 80: B is a point on the ray AA, then r
Page 81 and 82: We shall use these concepts to prov
Page 83 and 84: using reflections. let ABC be a tri
Page 85 and 86: are: Translations. This is the case
Page 87 and 88: y the equation (w , w 1 ;w 2 ,w 3 )
Page 89 and 90: is a plane (which, of course, remai
Page 91 and 92: Proposition 38 If ABCD is a tetrahe
Page 93 and 94: • f(x 1 ) = −x 1 ad A is the ma
Page 95 and 96: The cube: The eight points A (1,−
Page 97 and 98: • there is an n ∈ N so that G n
Page 99 and 100: Then one can verify that phi 1 and
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We remark that we have natural mapp
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V. In general, a group action on a
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this representation is unique i.e.
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Note that if m r is the order of th
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If R is a ring with unit, then not
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If R is commutative, we can define
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As noted above, we need not have in
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Proof. Let I be an ideal in R which
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As in the scalar case, these operat
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For the proof it suffices to show t
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where the b ij are in E. Then x =
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