Degree Complexity of a Family of Birational Maps

Bedford, Eric; Kim, Kyoung-Hee; Truong, Tuyen Trung; Abarenkova, N.; Maillard, J.‐M.

doi:10.1007/s11040-008-9039-6

Cited by 12 publications

(31 citation statements)

References 29 publications

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“…Then n = 3, a 0 = 2, and the map k F is exactly the family considered in Section 5 in [2]. Hence in this case δ(k F ) = 1.…”

Section: Introductionmentioning

confidence: 91%

“…We continue the work of [2] where we considered a family k F of birational maps of the plane determined by a choice of polynomial F (w) = a 0 + a 1 w + . .…”

Section: Introductionmentioning

confidence: 99%

“…+ a n w n (the definition of the family k F will be recalled in Section 2). In [2] we determined the degree complexity δ(k F ) in the generic case. If δ(k b F ) is less than the generic value, then we say that F is exceptional.…”

Section: Introductionmentioning

confidence: 99%

“…As seen in [2], there is a fundamental difference between the cases where n, the degree of F, is even or odd.…”

Section: Introductionmentioning

confidence: 99%

“…We compute the degree complexity of the family of birational maps considered in [2] for all exceptional cases. Some interesting properties of the family are also given.…”

mentioning

confidence: 99%

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Degree Complexity of a Family of Birational Maps: II. Exceptional Cases

Truong

2009

Math Phys Anal Geom

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Abstract. We compute the degree complexity of the family of birational maps considered in [2] for all exceptional cases. Some interesting properties of the family are also given. IntroductionWe continue the work of [2] where we considered a family k F of birational maps of the plane determined by a choice of polynomial F (w) = a 0 + a 1 w + . . . + a n w n (the definition of the family k F will be recalled in Section 2). In [2] we determined the degree complexity δ(k F ) in the generic case. If δ(k b F ) is less than the generic value, then we say that F is exceptional. (The set of exceptional parameters is a nowhere dense algebraic subset.) This corresponds to cases of degree reduction which are especially interesting because they correspond to the maps that have special symmetries.As seen in [2], there is a fundamental difference between the cases where n, the degree of F, is even or odd.The complexity degree δ(k F ) in the case n is even is given byCase 2: If a 0 = 2 1+m for some integer m ≥ 0 then δ(k F ) is the largest real root of the polynomial x 2m+1 (x 2 − (n + 1)x − 1) + x 2 + n.The complexity degree δ(k F ) in the case n is odd is given by 0 , a 1 , . . . , a n ) = −(a n−j + a n−j+1 ) − [−a n n j + a n−1with n! = n(n − 1) . . . 2.1 the factorial of n. a 0 , a 1 , . . . , a n ) = 0 for all 0 ≤ j ≤ h.Case 1: h < n − 2, and a 0 = 2/(1 + m) for all integers m ≥ 0. Then δ(k F ) is the largest real root of the polynomial x 3 − nx 2 − (n + 1 − h)x − 1.Case 2: h < n − 2, and a 0 = 2/(1 + m) for some integer m ≥ 0. Then δ(k F ) is the largest real root of the polynomial x 2m+1 (x 3 − nx 2 − (n − h + 1)x − 1) + x 3 + x 2 + nx + n − h − 1.Case 3: h = n − 2, and a 0 = 2/(1 + m) for all integers m ≥ 0, and a 0 = n+1 2 + l 2(1+l) for all integers l ≥ 0. Then δ(k F ) is the largest real root of the polynomial x 3 − nx 2 − 2x − 1.Case 4: h = n − 2, and a 0 = 2/(1 + m) for some integer m ≥ 0, and a 0 = n+1 2 + l 2(1+l) for all integers l ≥ 0. Then δ(k F ) is the largest real root of the polynomial x 2m (x 3 − nx 2 − 2x − 1) + x 2 + x + n.Case 5: h = n − 2, and a 0 = 2/(1 + m) for all integers m ≥ 0, and a 0 = n+1 2 + l 2(1+l) for some integer l ≥ 0. Then δ(k F ) is the largest real root of the polynomial x 2l+2 (x 3 − nx 2 − 2x − 1) + nx 2 + x + 1.Case 6: h = n − 2, and a 0 = 2/(1 + m) for some integer m ≥ 0, and a 0 = n+1 2 + l 2(1+l) for some integer l ≥ 0. Then n = 3, a 0 = 2, and the map k F is exactly the family considered in Section 5 in [2]. Hence in this case δ(k F ) = 1.

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“…Then n = 3, a 0 = 2, and the map k F is exactly the family considered in Section 5 in [2]. Hence in this case δ(k F ) = 1.…”

Section: Introductionmentioning

confidence: 91%

“…We continue the work of [2] where we considered a family k F of birational maps of the plane determined by a choice of polynomial F (w) = a 0 + a 1 w + . .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As seen in [2], there is a fundamental difference between the cases where n, the degree of F, is even or odd.…”

Section: Introductionmentioning

confidence: 99%

“…We compute the degree complexity of the family of birational maps considered in [2] for all exceptional cases. Some interesting properties of the family are also given.…”

mentioning

confidence: 99%

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Degree Complexity of a Family of Birational Maps: II. Exceptional Cases

Truong

2009

Math Phys Anal Geom

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Continuous families of rational surface automorphisms with positive entropy

Bedford

Kim

2010

Math. Ann.

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Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space

Silverman

2012

Ergod. Th. Dynam. Sys.

114

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Let ϕ : P N P N be a dominant rational map. The dynamical degree of ϕ is the quantity δ ϕ = lim(deg ϕ n ) 1/n . When ϕ is defined overQ, we define the arithmetic degree of a point P ∈ P N (Q) to be α ϕ (P) = lim sup h(ϕ n (P)) 1/n and the canonical height of P to beĥ ϕ (P) = lim sup δ −n ϕ n − ϕ h(ϕ n (P)) for an appropriately chosen ϕ . We begin by proving some elementary relations and making some deep conjectures relating δ ϕ , α ϕ (P), h ϕ (P), and the Zariski density of the orbit O ϕ (P) of P. We then prove our conjectures for monomial maps.

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Degree Complexity of a Family of Birational Maps

Abstract: We compute the degree complexity of a family of birational mappings of the plane with high order singularities

Cited by 12 publications

References 29 publications

Degree Complexity of a Family of Birational Maps: II. Exceptional Cases

Degree Complexity of a Family of Birational Maps: II. Exceptional Cases

Continuous families of rational surface automorphisms with positive entropy

Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space

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