熱帶亞純函數及其在差分方程之應用 - 政大學術集成

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(1)國立政治大學應用數學系碩士學位論文指導教授：. 蔡炎龍. 博士. 政治大. ‧. ‧ 國. 學. 立 Tropical Meromorphic Functions and Their Application on Difference Equations 熱帶亞純函數及其在差分方程之應用 n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 碩士班學生：陳亮撰中華民國一百年七月一日.

(2) 誌. 謝. 時間過得真快，轉眼間在政大待了三年了。有點不敢相信自己可以準時畢業，記得三月的時候還在擔心論文寫不出來，找指導教授蔡炎龍老師問說如果論文來不及寫完，能不能帶論文實習。幸好蔡老師有一篇熱帶幾何的論文可以套用Latex格式，也幸好之前碩一時有修過蔡老師開的數學程式設計專題，從中學到了如何用Latex打論文，還有beamer簡報系統，動態幾何軟體Geogebra，這些在我論文生成的過程中全部都派上了用場，此外也學到了Python，Flex等等，這麼多的幸好讓我能夠準時的完成這篇論文。感謝蔡炎龍老師帶我進入熱帶幾何這個有趣的新領域，雖然自己只能窺得一小部份，感謝陳天進老師在之前meeting還有口試時提供寶貴的意見，感謝台大數學系的助理教授劉瓊如博士特地前來參加我的論文口試。. 立. 政治大. ‧. ‧ 國. 學. 這三年在政大過得很充實，有許多美好的回憶。一二年級時住在自強九舍D221，剛好同學陳家盛住在隔壁D222，一起拼實變期中期末考，實變資格考，一起跑步，吃大餐，唱歌，到處遊玩· · · · · · ，感謝這三年能夠認識你，讓碩士生活變得更精彩，但唯一遺憾的就是這三年我還是交不到女朋友。政大應數所有一個很棒的研究室，整個和樂融融，常常會有一些聚餐，活動等，增進師生間以及學長姐學弟妹間的感情。山上宿舍環境清幽，空氣新鮮；校內一元公車交通便利，司機親切有人情。畢業在即，卻有幾分不捨。. sit. y. Nat. n. al. er. io. 相對於大學時自己的茫然不上進，研究所這三年比較知道要做什麼，不僅修完所上規定的30學分，還另外修習了教育學程，而且這三年來都有在補習班打工或是家教，所得足夠支付學雜費及自己的生活所需。碩一時接了蔡老師財管系大一微積分助教，碩三時接了陳老師的高微助教，由於自己的高微底子不是很好，幾乎每個星期二晚上都準備演習課到早上，感謝盈穎一起搭檔高微助教，分擔了許多工作。雖然辛苦，但撐過之後覺得自己的分析能力提升了不少，對於論文的撰寫有很大的助益。. Ch. engchi. i n U. v. 最後再次感謝三位口試老師在口試期間給予的意見與補充，才能讓本論文更加完善。也感謝家人支持我在當完兵之後繼續回學校就讀研究所，之後8月要在板橋高中實習半年，接著考教檢教甄，希望可以一切順遂。陳亮. 謹誌于. 國立政治大學應用數學所中華民國一百年七月.

(3) Abstract 治政大 In this thesis, we find the formula of tropical meromorphic function by giving finite number of立 roots and poles (with multiplicities). We also find. ‧. ‧ 國. 學. a simple formula for tropical periodic function by giving finite number of roots and poles (with multiplicities) during a period [0, T ). We then discuss all cases of the tropical meromorphic solution functions of first-order linear difference equation. At last, we provide a tropical approximated function of a given continuous function. We hope it is helpful in solving the tropical meromorphic solution functions of a given difference equation.. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(4) 政摘要治大在這篇論文中，一個熱帶亞純函數(tropical meromorphic function)若給定立有限個零根(roots)與極點(poles)還有它們的重數(multiplicities)，我們證明了這 ‧. ‧ 國. 學. 個熱帶亞純函數的存在與唯一性。另外，一個熱帶週期函數(tropical periodic function)若給定一個週期區間內的有限個零根與極點還有它們的重數，我們也找到了這個熱帶週期函數的一個簡單表達式。接著，給定一個一階線性差分方程(first-order linear difference equation)，我們討論了各種情況下的所有熱帶亞純函數解的表達式。最後，對於連續函數我們提供了一個它的熱帶近似函數，希望對於解差分方程的熱帶亞純函數解時能有所助益。. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(5) Contents 1 Introduction. 政治大. 2. 2 Background 2.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Equalities and Inequalities . . . . . . . . . . . . . . . . .. 立. ‧ 國. 學. 4 4 5. ‧. 3 Tropical Meromorphic Functions 7 3.1 Tropical Polynomials in One Variable . . . . . . . . . . . . . . 8 3.2 Tropical Meromorphic Functions . . . . . . . . . . . . . . . . . 10 3.3 Extended Tropical Meromorphic Functions . . . . . . . . . . . 18. y. Nat. sit. n. al. er. io. 4 Tropical Periodic Functions 22 4.1 Generating Functions of Tropical Periodic Functions . . . . . 23 4.2 R-linear Combination of Tropical 1-Periodic Functions . . . . 25 4.3 R-linear Combination of Tropical T -Periodic Functions . . . . 29. Ch. engchi. i n U. v. 5 Application on Difference Equations 5.1 Tropical Counterpart to The Exponential Function . . . . . . 5.2 Application on Difference Equations: First Order . . . . . . . 5.3 Tropical Approximated Function . . . . . . . . . . . . . . . .. 1. 35 36 38 42.

(6) Chapter 1 Introduction 立. 政治大. ‧. ‧ 國. 學. Tropical geometry is a relatively new area in mathematics, first developed in the 1980s by Imre Simon, a mathematician and computer scientist from Brazil. It is a piecewise-linear version of algebraic geometry, the algebraic structure we work on is the set T = R ∪ {−∞} equipped with addition and multiplication defined by:. y. sit. Nat. x ⊕ y = max{x, y}, x

(7) y = x + y.. n. al. er. io. In words, the tropical sum of two numbers is their maximum, and the tropical product of two numbers is their usual sum. These two operations both are associative and commutative, and the multiplication is distributive with respect to the addition. The identity elements for the tropical operations are 0T = −∞ for addition and 1T = 0 for multiplication. Observe that such a structure is not a ring, since not all elements have tropical additive inverses. For example, the function x ⊕ 1 = 2 has no solutions in T. We can define the tropical semiring in another way. For example, x ⊕ y = min{x, y}, x

(8) y = x + y. This is min-plus tropical semiring. However, the work on min-plus tropical semiring is similar to the work on max-plus tropical semiring. We will work on max-plus tropical semiring in this thesis, and only with one variable. For basic tropical geometry, one can see book [5], and [1], [2], [6], [7]. In [1] and [2], Grigg and Manwaring give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring,. Ch. engchi. 2. i n U. v.

(9) and they provide a simple algorithm for factoring tropical polynomials of a single variable. The main references of this thesis is [3] and [4], both are introducing tropical nevenlinna theory and applications on ultra-discrete equations, which is an equation, written in terms of addition and max operators, such that both dependent and independent variables take only discrete values. Here we do not mention the tropical nevenlinna theory, and emphasizing on tropical meromorphic function with integer slope or real slope (extended). Given finite number of roots and poles (with multiplicities respectively) of a tropical meromorphic function, we can represent the function immediately just like general rational funtion by giving roots and poles (Chapter 3). And in Chapter 4, we define a function. 立. 政治大. πa (x) := max{(1 − a)([x] − x), a([−x] − (−x))}.. ‧ 國. 學. ‧. to generate the tropical T -periodic functions by giving roots and poles (with multiplicities respectively) during an period [0, T ). In chapter 5, we want to obtain a representation of all the extended tropical meromorphic solution functions of first-order linear difference equation y(x + 1) = ay(x) + b, for a, b ∈ R. At last, we provide a tropical approximated function of a given continuous function f (x) by. io. sit. y. Nat. n. al. er. (f ([x + 1]) − f ([x]))(x − [x]) + f ([x]).. Ch. i n U. v. It may be helpful in dealing with ultra-discrete equations, at least for firstorder linear difference equations.. engchi. 3.

(10) Chapter 2 Background 立 Arithmetic. 學. ‧ 國. 2.1. 政治大. Definition 2.1. We define operators ⊕ : T × T −→ T, and

(11) : T × T −→ T by:. ‧. x ⊕ y := max{x, y}, x

(12) y := x + y.. y. Nat. sit. n. al. er. io. Remark 2.1. For each x ∈ T, x ⊕ (−∞) = −∞, hence −∞ is the additive identity element. x

(13) 0 = x, hence 0 is the multiplicative identity element. Example 2.1.. Ch. engchi. i n U. v. 2 ⊕ 3 = max{2, 3} = 3; 2 ⊕ 2 = max{2, 2} = 2; 2 ⊕ (−∞) = max{2, −∞} = 2. 2

(14) 3 = 2 + 3 = 5; 2

(15) 0 = 2 + 0 = 2; 2

(16) (−∞) = 2 + (−∞) = −∞. Definition 2.2. For each integer n, define x

(17) n := n × x. And define tropical division to be their usual subtraction: x y := x − y.. 4.

(18) Moreover, define n M i=1 n K. ai := max{a1 , a2 , · · · , an } ai := a1 + a2 + · · · + an =. i=1. n X. ak. k=1. The definition of semiring are given formally as the following:. 政治大. Definition 2.3 (Semiring). A semiring is a set S equipped with two algebraic operations, called addition and multiplication, such that:. 立. (i) The addition and multiplicatioon are associative.. ‧ 國. 學. (ii) The addition is commutative.. (iii) The multiplication is distributive with respect to the addition.. er. io. sit. y. Nat. al v Some Equalities and Inequalities ni n. 2.2. ‧. Remark 2.2. (R ∪ {−∞}, ⊕,

(19) ) is a semiring.. Ch. engchi U. Under these new operations, there are some funny equalities and inequalities. (i) (b a) ⊕ (d c) = (b

(20) c ⊕ a

(21) d) (a

(22) c) It is an analogue of bc + ad b d + = . a c ac (ii) x

(23) (1T n) = 1T (x

(24) n ) = −n × x It is an analogue of 1 1 xn = n . x. 5.

(25) (iii) The Freshman’s dream come true in tropical arithmetic: (x ⊕ y)

(26) n = x

(27) n ⊕ y

(28) n (iv) Tropical Cauchy inequality holds: n n n M M M

(29) 2 ( a

(30) 2 )

(31) ( b ) ≥ ( (ai

(32) bi ))

(33) 2 i i i=1. i=1. i=1. i.e.. 政治大. max{a1 , · · · , an } + max{b1 , · · · , bn } ≥ max{a1 + b1 , · · · , an + bn } (2.1). 立. (v) for aij ∈ R ∪ {−∞}, i = 1, 2, · · · , m; j = 1, 2, · · · , n,. i=1. i=1. j=1. M. (2.2). i=1 j=1. aij. 1≤i≤n,1≤j≤m. io. n. er. j=1 i=1. X. sit. Nat. n X m m X n X X ( aij ) = ( aij ) =. y. It is an analogue of. al. aij. 1≤i≤m,1≤j≤n. ‧. ‧ 國. j=1. 學. n M m m M n M M ( aij ) = ( aij ) =. v. Proof. (iv): Let ar = max{a1 , a2 , · · · , an }, bs = max{b1 , b2 , · · · , bn }, then ar ≥ ak for each k = 1, 2, · · · , n. And bs ≥ bk for each k = 1, 2, · · · , n. Hence, ar + bs ≥ ak + bk for each k = 1, 2, · · · , n. Thus, (2.1) holds.. Ch. engchi. i n U. Remark 2.3. Let A = {r|ar = max{a1 , a2 , · · · , an }}, B = {s|bs = max{b1 , b2 , · · · , bn }}, then the equal sign of (2.1) holds if and only if A ∩ B 6= φ.. 6.

(34) Chapter 3 Tropical Meromorphic Functions 政治大立 ‧ 國. 學. ‧. In this Chapter, we will derive the equation of tropical meromorphic functions with prescribed roots and poles. In section 3.1, we first introduce the tropical polynomials in one variable, and then tropical meromorphic functions in section 3.2, if the slope is allowed to be non-integer, it is extended tropical meromorphic functions in section 3.3.. n. er. io. sit. y. Nat. al. Ch. engchi. 7. i n U. v.

(35) 3.1. Tropical Polynomials in One Variable. Definition 3.1. In general, the tropical polynomials in one variable are n M f (x) = (ai

(36) x

(37) i ) = an

(38) x

(39) n ⊕ an−1

(40) x

(41) (n−1) ⊕ · · · · · · ⊕ a1

(42) x ⊕ a0 . i=0. Here the coefficients an , an−1 , · · · · · · , a0 are real numbers and n is a nonnegative integer. When evaluating this function in classical arithmetic, we have f (x) = max{an + nx, an−1 + (n − 1)x, · · · · · · , a1 + x, a0 }.. 政治大. Remark 3.1. It is clear that this function f (x) maps T to T has the following three important properties:. 立. (i) f (x) is continuous.. ‧ 國. 學. (ii) f (x) is piecewise-linear, where the number of pieces is finite. 1 x+y ) ≤ (f (x) + f (y)) for all x, y ∈ R. 2 2. ‧. (iii) f (x) is convex, i.e., f (. n. al. er. io. sit. y. Nat. Example 3.1. f (x) = x

(43) 2 ⊕ 1 is a tropical polynomial, the coefficient of x

(44) 2 is 0, and the coefficient of x is −∞, it is a missing term, and the constant 1 term is 1. We will see f (x) = max{2x, 1}, i.e., f (x) = 1 if x ≤ ; and 2 1 f (x) = 2x if x > . (Figure 3.1(a)) 2 1 Example 3.2. g(x) = x

(45) 2 ⊕

(46) x⊕1 (Figure 3.1(b)), h(x) = x

(47) 2 ⊕(−1)

(48) x⊕1 2 (Figure 3.1(c)). From the graph, we see f (x) in Example 3.1 and g(x), h(x) are different tropical polynomials which define the same function. And p(x) = x

(49) 2 ⊕ 1

(50) x ⊕ 1 (Figure 3.1(d)) defines a different function of them. 1 From Figure 3.1, we know that if the coefficient of x is less or equal to , then 2 1 they all define the same function; and if the coefficient of x is larger than , 2 1 then it will be a different function, it seems that is the largest coefficient 2 of x to unchange the function. So we have the following definition.. Ch. engchi. i n U. v. Definition 3.2. Given two polynomials g(x) and h(x), if g(c) = h(c) for all c ∈ R, then g(x) and h(x) are functionally equivalent. 8.

(51) (a) f (x) = x

(52) 2 ⊕ 1.. 立. (b) g(x) = x

(53) 2 ⊕. 1

(54) x ⊕ 1. 2. 政治大. ‧. ‧ 國. 學. (c) h(x) = x

(55) 2 ⊕ (−1)

(56) x ⊕ 1.. Nat. Figure 3.1:. n. al. er. io. sit. y. (d) p(x) = x

(57) 2 ⊕ 1

(58) x ⊕ 1.. v. In classical, if g(x) and h(x) are functionally equivalent, if and only if g(x) and h(x) are the same polynomials. But in tropical geometry, functional equivalence does not mean the functions obtained from the same polynomials. Although different tropical polynomials might define the same function, we can adjust each of coefficient to be maximal, and to get a maximally represented tropical polynomial which still represent the same function. One can see section 3 in [8] to know how to derive the maximally represented tropical polynomial of a given tropical polynomial. And the similar result can be found in [2] and [1] on min-plus tropical polynomial, it is call leastcoefficient tropical polynomial in [2] and [1]. Moreover, each maximally represented tropical polynomial (or least-coefficient tropical polynomial on min-plus tropical polynomial) can be factorized into a product of linear terms, it is the Fundamental Theorem of Tropical Algebra, you can see the proof in [2] on min-plus form.. Ch. engchi. 9. i n U.

(59) 3.2. Tropical Meromorphic Functions. Definition 3.3. A tropical rational function is a function of the form R(x) = (am

(60) x

(61) m ⊕ · · · ⊕ a1

(62) x ⊕ a0 ) (bn

(63) x

(64) n ⊕ · · · ⊕ b1

(65) x ⊕ b0 ) where m and n are non-negative integers,and ai , bj ∈ T, i = 1, 2, · · · , m; j = 1, 2, · · · , n. Definition 3.4. A continuous piecewise linear function f : R → R is said to be a tropical meromorphic function on R if both one-sided derivatives are integers at each point x ∈ R.. 治政 In this thesis, if f (x) is a tropical meromorphic大 function on R, we call f (x) 立 function. Note that every tropical polynomial is an R-tropical meromorphic ‧ 國. 學. an R-tropical meromorphic function, and so is tropical rational function. Moreover, the word “piecewise” implies that there are finitely many pieces in every bounded interval.. Nat. y. ‧. Example 3.3. f (x) = (x

(66) 2 ⊕ 0) x (Figure 3.2(a)) is a tropical rational function. In fact, it is equal to |x|. It shows in the following.. n. al. where x+ = max{x, 0}, x− = max{−x, 0}. Summation of the above two equalities, it gets. Ch. engchi. er. io. x = x+ − x−. sit. |x| = x+ + x−. i n U. v. |x| = 2x − x = 2 max{x, 0} − x = max{2x, 0} − x = (x

(67) 2 ⊕ 0) x +. Example 3.4. Figure 3.2(b) shows a function which is an R-tropical meromorphic function but not a rational function. Note that in tropical functions, those non-differentiable points are special, it comes the following definitions. Definition 3.5. For each x ∈ R, let wf (x) = lim+ (f 0 (x + ) − f 0 (x − )) →0. 10.

(68) (b) f (x) = |x|, −1 ≤ x ≤ 1; f (x + 治 for each x ∈ R.. 政 2) = f (x) 大. (a) f (x) = (x

(69) 2 ⊕ 0) x = |x|.. 立. Figure 3.2:. ‧ 國. 學. (i) If wf (x) > 0, then x is called a root of f with multiplicity wf (x).. ‧. (ii) If wf (x) < 0, then x is called a pole of f with multiplicity −wf (x).. sit. y. Nat. (iii) If wf (x) = 0, then x is called an ordinary point of f .. io. er. Note that all roots and poles of f constitute the support of wf (x). Consider the function f (x) = x ⊕ a = max{x, a}, which has a root at a. From the point of view of algebra and factorization, it is perhaps more natural to think of a as the negative of the root. However, there is no subtraction in the tropical semi-ring and the definition given above is the natrual one from the point of view of geometry.. n. al. Ch. engchi. i n U. v. Definition 3.6. If f is tropical meromorphic on R and f 0 (x) = m for all x < x0 , for some constant m ∈ Z and x0 ∈ R, then we say that f is tropical meromorphic on T. Define wf (−∞) = m and (i) If m > 0, then the point −∞ is called a root of order m. (ii) If m < 0, then the point −∞ is called a pole of order −m. (iii) If m = 0, then the point −∞ is called an ordinary point.. 11.

(70) If f (x) is a tropical meromorphic function on T, we call f (x) an T-tropical meromorphic function. We know that in traditional rational functions, given roots and poles and their multiplicities, we can find a suitable rational functions. Example 3.5. Given two roots −1 and −2 with multiplicities 2 and 5 respectively, and one pole −3 with multiplicity 7, and no other roots or poles. Then k(x + 1)2 (x + 2)5 , k is a nonzero constant. the suitable function is f (x) = (x + 3)7 We have a similar tropical result in the following. Before it, we need some lemmas at first.. 治政大 f (x) are R-tropical Lemma 3.1 (The Linear Property of w (x)). Suppose meromorphic functions,立 i = 1, 2, · · · , n, and c , c , · · · , c are constants in f. i. f (x) =. n X. 2. n. 學. ci fi (x),. i=1. wf (x) =. n X. ‧. ci wfi (x). Nat. y. i=1. sit. then. ‧ 國. 1. R. Let. n. al. er. io. Proof. If x is not a root or pole of any fi (x), then f 0 (x) = hence, for any x ∈ R, and small ,. i n U. wf (x) = lim(f 0 (x + ) + f 0 (x − )) →0. Ch. engchi. v. n n X X = lim( ci fi0 (x + ) − ci fi0 (x − )) →0. i=1. = lim( →0. = =. n X i=1 n X. n X. i=1. ci (fi0 (x + ) − fi0 (x − ))). i=1. ci lim(fi0 (x + ) − fi0 (x − )) →0. ci wfi (x). i=1. 12. Pn. 0 i=1 ci fi (x),.

(71) Lemma 3.2. Two R-tropical meromorphic functions f (x) and g(x) satisfy the relation wf (x) − wg (x) ≡ 0 if and only if f (x) − g(x) is a linear function on R. That is, f (x) − g(x) = Ax + B for some constants A and B. Proof. Since f (x) and g(x) are R-tropical meromorphic functions, let h(x) = f (x) − g(x), then by Lemma 3.1, wh (x) = wf (x) − wg (x) = 0 Therefore, h0 (x+) = h0 (x−) holds for any x ∈ R. Then wh (x) ≡ 0 implies h0 (x) is a constant on R. Hence, h(x) = f (x) − g(x) is a linear function on R. Coversely, of course that the w of any linear function vanishes identically.. 政治大. 立. ‧. ‧ 國. 學. Theorem 3.3. Given c1 < c2 < · · · < cn ∈ T, and α1 , α2 , · · · , αn ∈ Z, there exist a tropical rational function f such that wf (ci ) = αi for i = 1, 2, · · · , n and wf (x) = 0 for x 6= ci , i = 1, 2, · · · , n.(i.e., ci is a root if αi > 0 and a pole if αi < 0, and for any x 6= c1 , c2 , · · · , cn is an ordinary point.) In fact, n K f (x) = k

(72) ( (x ⊕ ci )

(73) αi ). y. Nat. i=1. sit. er. io. with k a constant in R is the unique function satisfying the condition, the uniqueness is up to the constant k.(Note that c1 might be −∞.) First prove the existence of f . Define m0 = 0 and. n. al. Ch. engchi. i n U. mj = α1 + α2 + · · · + αj =. j X. v. αi. i=1. to be the slope of f (x) in (cj , cj+1 ), then n K f (x) = k

(74) ( (x ⊕ ci )

(75) αi ) i=1. =k+. n X. αi max{x, ci }. i=1. =k+. n X. (mi − mi−1 ) max{x, ci }. i=1. 13.

(76) 治政 Figure 3.3: 大. Proof.. 立. (mi − mi−1 ) max{x, ci } +. i=1. (mi − mi−1 )ci. er. (mi − mi−1 )ci. n. al. sit. i=j+1 n X. io. = k + mj x +. n X. y. (mi − mi−1 )x +. i=1. i=j+1. Therefore,. (mi − mi−1 ) max{x, ci }. i=j+1. Nat. =k+. j X. n X. ‧. ‧ 國. f (x) = k +. j X. 學. If x ∈ (cj , cj+1 ) for some j = 1, 2, · · · , n, then. Ch. f 0 (x) = mj. engchi f or. i n U. v. x ∈ (cj , cj+1 ).. If x = ci for some i = 2, 3, · · · , n, then wf (ci ) = lim+ (f 0 (ci + ) − f 0 (ci − )) →0. = lim+ f 0 (ci + ) − lim+ f 0 (ci − ) →0. →0. Since ci + ∈ (ci , ci+1 ) and ci − ∈ (ci−1 , ci ), by (3.1) it follows wf (ci ) = mi − mi−1 = αi 14. (3.1).

(77) If c1 6= −∞, let c0 = −∞ ,−∞ is a ordinary point implies that f 0 (x) = 0 for all x < x0 for some x0 ∈ R. Since there is no other non-ordinary point between c0 and c1 , so f 0 (x) = 0 f or. all. x < c1 .. (3.2). Hence, wf (c1 ) = lim+ (f 0 (ci + ) − f 0 (ci − )) = m1 − 0 = α1 →0. If c1 = −∞, by (3.1) it follows f or x ∈ (c , c ) 政治大. f 0 (x) = m1. 立. i.e.,. ‧ 國. f or. all. 2. x < c2 ,. 學. then. f 0 (x) = m1. 1. ‧. wf (−∞) = m1 = α1 wf (c1 ) = α1. y. Nat. n. er. io. sit. It completes the proof of existence.. al. (3.3). Ch. i n U. v. Now prove the uniqueness of f . If g(x) is another function satisfying wg (ci ) = αi , i = 1, 2, · · · , n; and wg (x) = 0 for any x 6= c1 , c2 , · · · , cn . Then we will prove the uniqueness by proving that f (x) − g(x) is a constant.. engchi. case 1: c1 6= −∞ On this case,the least non-ordinary point is c1 , hence c0 = −∞ is an ordinary point, then wg (−∞)=0, it follows g 0 (x) = 0 for all x < c1 , and by (3.2) we know f 0 (x) = 0 for all x < c1 . Let h(x) = f (x) − g(x), then h0 (x) = f 0 (x) − g 0 (x) = 0 − 0 = 0. 15. f or. x < c1 .. (3.4).

(78) And by Lemma 3.1, wh (x) = wf (x) − wg (x) ( 0−0=0 = wf (ci ) − wg (ci ) = 0. if x 6= c1 , c2 , · · · , cn if x = ci for some i = 1, 2, · · · , n.. Lemma 3.2 implies h(x) = Ax+B, for some constant A and B. And equation (3.4) implies A = 0, therefore, h(x) = f (x) − g(x) is a constant.. case 2: c1. 治政大 = −∞. (i.e., w (−∞) = α ) 立 g. 1. ‧. ‧ 國. 學. It follows g 0 (x) = α1 for all x < c2 , and by (3.3), f 0 (x) = α1 for all x < c2 . Then h0 (x) = f 0 (x) − g 0 (x) = α1 − α1 = 0 for all x < c2 . Hence, wh (c1 ) = 0, and h(x) is a constant for all x < c2 . If x = ci for i = 1, 2, · · · , n, then wh (ci ) = wf (ci ) − wg (ci ) = αi − αi = 0. y. Nat. n. wh (x) = 0. Ch. holds for the same reason in case 1 too. It completes the proof of uniqueness.. engchi. er. io. al. sit. holds for the same reason in case 1. And if x is an ordinary point,. i n U. v. It follows the next theorem immediately. Theorem 3.4. If a function is R-tropical meromorphic, then it is tropical rational if and only if it has a finite number of roots and poles. Proof. By definition it is clear that if f is tropical rational then it has a finite number of roots and poles; conversely, if a tropical meromorphic function f has a finite number of roots and poles then f is a tropical rational function by Theorem 3.3. So the Figure 3.2(b) in Example 3.3 with infinite roots and poles implies that it is not a tropical rational function. 16.

(79) 1 of multiplicities 2. Example 3.6. Let g(x) be a polynomial with root 2 Applying Theorem 3.3, one solution is the following polynomial: 1 1 1 g(x) = (x ⊕ )

(80) 2 = (x ⊕ )

(81) (x ⊕ ) 2 2 2 1 1 1 1 = x

(82) 2 ⊕ x

(83) ⊕

(84) x ⊕

(85) 2 2 2 2 1 = x

(86) 2 ⊕

(87) x ⊕ 1 2 We find that it is exactly the function in Figure 3.1(b). And since max{2x, 1} =. 立. then. 政治大. max{2x, 1} + max{2x, 1} 2x + 1 1 ≥ =x+ , 2 2 2 g(x) = x

(88) 2 ⊕ 1. ‧ 國. 學. ‧. 1 But x

(89) 2 ⊕

(90) x ⊕ 1 is the maximally represented tropical polynomial of g(x). 2 Example 3.7. f1 (x) = (x ⊕ (−2))

(91) (x ⊕ (−1))

(92) 2

(93) (x ⊕ 1)

(94) 3 is a tropical polynomial with roots −2, −1, 1 of multiplicities 1, 2, 3 respectively.(Figure 3.4(a)). f2 (x) = (x)

(95) (x ⊕ (−1))

(96) 2

(97) (x ⊕ 1)

(98) 3 is a tropical polynomial with roots −∞, −1, 1 of multiplicities 1, 2, 3 respectively. (Figure 3.4(b)). sit. y. Nat. n. al. er. io. Example 3.8. In Example 3.3, we have show that |x| is a tropical rational function, now we can get the expression of |x| in tropical rational form immediately. From the graph of |x|, we know |x| have a pole −∞ of multiplicity 1, and a root 0 of multiplicity 2. Therefore, |x| = k

(99) (x ⊕ 0)

(100) 2 (x ⊕ (−∞)) for some constant k, k = 0 follows from |0| = 0, one can omit −∞ to get the same expression. Ch. engchi. i n U. v. |x| = (x ⊕ 0)

(101) 2 x = (x

(102) 2 ⊕ 0) x Example 3.9. Find the tropical rational function f (x) with non-ordinary points −∞, −1, 0, 1, and wf (−∞) = 1, wf (−1) = −2, wf (0) = 2, wf (1) = −2. That is, −∞, 0 are roots of multiplicities 1, 2 respectively; and −1, 1 are poles of multiplicities 2, 2 respectively. Therefore, f (x) = k

(103) (x ⊕ (−∞))

(104) 1

(105) (x ⊕ (−1))

(106) (−2)

(107) (x ⊕ 0)

(108) 2

(109) (x ⊕ 1)

(110) (−2) = k

(111) (x ⊕ (−∞))

(112) (x ⊕ 0)

(113) 2 ((x ⊕ (−1))

(114) 2

(115) (x ⊕ 1)

(116) 2 ) = k

(117) (x)

(118) (x ⊕ 0)

(119) 2 ((x ⊕ (−1))

(120) 2

(121) (x ⊕ 1)

(122) 2 ) 17.

(123) 政治大. 立. (b) f2 (x).. 學. ‧ 國. (a) f1 (x).. Figure 3.4:. ‧. If we hope f (0) = 0, then k = 2.. er. io. sit. y. Nat. f (x) = 2

(124) (x)

(125) (x ⊕ 0)

(126) 2 ((x ⊕ (−1))

(127) 2

(128) (x ⊕ 1)

(129) 2 )(F igure3.5(a)). Example 3.10. Given Figure 3.5(b), find the tropical rational expression g(x) of it. Since g 0 (x) = 0 for all x < −2, then −∞ is an ordinary point, the slope change at −2, −1, 1, 2, and wg (−2) = 2−0 = 2, wg (−1) = −2−2 = −4, wg (1) = 2 − (−2) = 4, wg (2) = 0 − 2 = −2.Then. n. al. Ch. engchi. i n U. v. g(x) = k

(130) (x ⊕ (−2))

(131) 2

(132) (x ⊕ (−1))

(133) (−4)

(134) (x ⊕ 1)

(135) 4

(136) (x ⊕ 2)

(137) (−2) = (k

(138) (x ⊕ (−2))

(139) 2

(140) (x ⊕ 1)

(141) 4 ) ((x ⊕ (−1))

(142) 4

(143) (x ⊕ 2)

(144) 2 ) k = 2 follows from g(0) = 2.. 3.3. Extended Tropical Meromorphic Functions. If we allow a tropical meromorphic function to have real slopes, this kind of tropical meromorphic function is called an extended tropical meromorphic function. So we have the following definitions. 18.

(145) (a) f (x) = 2

(146) (x)

(147) (x⊕0)

(148) 2 ((x⊕ (−1))

(149) 2

(150) (x ⊕ 1)

(151) 2 ).. 立. (b) g(x).. 政治大 Figure 3.5:. ‧ 國. 學. Definition 3.7. An extended tropical polynomial is of the form: f (x) = an

(152) x

(153) rn ⊕ an−1

(154) x

(155) rn−1 ⊕ a1

(156) x

(157) r1 ⊕ a0 ,. ‧. where ri ∈ R+ for all i = 1, 2, . . . , n; ai ∈ T, i = 0, 1, 2, · · · , n.. y. Nat. sit. Definition 3.8. An extended tropical rational function is of the form:. n. al. er. io. f (x) = an

(158) x

(159) rn ⊕ an−1

(160) x

(161) rn−1 ⊕ a1

(162) x

(163) r1 ⊕ a0 ,. i n U. v. where ri ∈ R \ {0} for all i = 1, 2, . . . , n; ai ∈ T, i = 0, 1, 2, · · · , n.. Ch. engchi. Definition 3.9. A continuous piecewise linear function f : R → R is said to be a extended tropical meromorphic function on R if both one-sided derivatives are real at each point x ∈ R Theorem 3.3 will hold too in the extended tropical rational functions, we restate the theorem in the following after slight modifications. Theorem 3.5. Given c1 < c2 < · · · < cn ∈ T, and α1 , α2 , · · · , αn ∈ R, there exist a extended tropical rational function f such that wf (ci ) = αi for i = 1, 2, · · · , n and wf (x) = 0 for x 6= ci , i = 1, 2, · · · , n.(i.e., ci is a root if αi > 0 and a pole if αi < 0, and for any x 6= c1 , c2 , · · · , cn is an ordinary point.). 19.

(164) In fact, n K f (x) = k

(165) ( (x ⊕ ci )

(166) αi ) i=1. with k a constant in R is the unique function satisfying the condition, the uniqueness is up to the constant k.(Note that c1 may be −∞.). 立. 政治大. Figure 3.6: h(x) = 1

(167) (x ⊕ (−∞)). 1 5 1 )

(168)

(169) (− ) 2

(170) (x ⊕ 0) 6

(171) (x ⊕ 3) 3. ‧. ‧ 國. 學.

(172) (−. sit. y. Nat. n. al. er. io. Example 3.11. Find the extended tropical rational expression h(x) of Fig1 ure 3.6. Since h0 (x) = − for all x < 0, then −∞ is a pole with wh (−∞) = 2 1 1 1 5 1 1 − , and wh (0) = − (− ) = , wh (3) = 0 − = − . Therefore, 2 3 2 6 3 3. Ch. engchi.

(173) (−. h(x) = k

(174) (x ⊕ (−∞)). i n U. v. 1 5 1 )

(175)

(176) (− ) 2

(177) (x ⊕ 0) 6

(178) (x ⊕ 3) 3. k = 1 follows from h(0) = 0. we check this is really the true function of Figure 3.6 in the following.. 20.

(179)  1 5 1   1 + (− )x + × 0 + (− )×3   2 6 3  5 1 1 h(x) = 1 + (− )x + × x + (− ) × 3  2 6 3   1 5 1   1 + (− )x + × x + (− ) × x 2 6 3  1   x≤0 − ×x   2 = 1 ×x 0≤x≤3   3  1 x≥3. x≤0 0≤x≤3 x≥3. 政治大 This is exactly the function of Figure 3.6. 立 ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 21. i n U. v.

(180) Chapter 4 Tropical Periodic Functions 政治大 In this chapter, we 立 examine the extended tropical periodic meromorphic ‧. ‧ 國. 學. functions. It will be shortened to be called tropical periodic function here. Given an extended tropical meromorphic function f on an interval [0, T ), with f (0) = limx→T − f (x), if we extend f to be a periodic function with period T, that is, f (x + T ) = f (x), can we find a formula to express it? It is “Yes”, let’s see how to do it in the following. In [4], Laine and Tohge consider a tropical meromorphic 1-period function defined by. sit. y. Nat. 1 max{a(x − [x]), −b((x − [x] − 1))} a+b 1

(181) = {(a(x − [x])) ⊕ (−b(x − [x]) − 1)} a + b. n. al. er. io. π (a,b) (x) =. Ch. engchi. i n U. v. for arbitrary parameters a, b ∈ R− . Laine and Tohge make conclusion that any non-constant tropical meromorphic 1-periodic function f (x) can be represented as an R-linear combination of such function π (a,b) (x). But here we provide a better function πa (x) = max{(1 − a)([x] − x), a([−x] − (−x))} = ([x] − x)

(182) (1−a) ⊕ ([−x] − (−x))

(183) a These functions will be used to generate the tropical periodic functions in section 4.2 and 4.3. We first prove some properties of πa (x) in section 4.1.. 22.

(184) 4.1. Generating Functions of Tropical Periodic Functions. Definition 4.1 (Tropical Periodic Function). If f (x) is an extended tropical meromorphic function on R, and there is a real number T > 0 such that f (x + T ) = f (x) for all x ∈ R, then we say f (x) is an extended tropical periodic meromorphic function. It will be shortened to be called tropical periodic function here. Let’s consider the following extended tropical meromorphic 1-periodic function. Definition 4.2. Define. 立. 政治大. πa (x) := max{(1 − a)([x] − x), a([−x] − (−x))}. ‧ 國. 學. for 0 ≤ a < 1. (Figure 4.1). ‧. The number a control the skewness of f (x). When a = 0, the graph is just the x-axis. It has the following properties.. y. n. al. (b). 2. sit. io. πa (n) = 0 f or each n ∈ Z.. (4.1). er. (a). Nat. Remark 4.1.. Ch. engchi. i n U. v. a − a = πa (a) ≤ πa (x) ≤ 0 f or 0 ≤ x < 1.. (4.2). (c) ( wπa (a) =. 1 0. a 6= 0 a=0. (4.3). (d) ( −1 F or n ∈ Z, wπa (n) = 0. 23. a 6= 0 a=0. (4.4).

(185) (a) a = 0.25.. 立. (b) a = 0.5.. 政治大. ‧ 國. 學 (c) a = 0.75.. ‧ sit er. io. (e). y. Nat. Figure 4.1: πa (x) = max{(1 − a)([x] − x), a([−x] − (−x))}. wπa (x) = 0 if x is not an integer and x 6= a.. v ni. n. al. Ch. U i e h n c g Proof. We verify these properties by a straightforward computation. (a) πa (n) = max{(1 − a)([n] − n), a([−n] − (−n))} = max{(1 − a) × 0, a × 0} = max{0, 0} = 0 (b) πa (a) = max{(1 − a)([a] − a), a([−a] − (−a))} = max{(1 − a)(−a), a(−1 + a)} = max{a2 − a, a2 − a} = a2 − a 24. (4.5).

(186) If 0 ≤ x < 1, πa (x) = max{(1 − a)(0 − x), a(−1 + x)} = max{ax − x, ax − a} ( (a − 1)x x≤a = ax − a x>a. (4.6) (4.7) (4.8). (a − 1)x ≤ 0 and ax − a = a(x − 1) ≤ 0, hence, πa (x) ≤ 0 for 0 ≤ x < 1. If x ≤ a, πa (x) = (a − 1)x ≥ (a − 1)a = a2 − a; if x > a, πa (x) = ax − a > a × a − a = a2 − a, hence, πa (x) ≥ a2 − a for 0 ≤ x < 1. That is, a2 − a is the minimum of πa (x).. 立. 政治大 (π (a + ) − π (a − )) = lim. ‧ 國. 學. 0 (c) By (4.8), wπa (a) = lim→0+ a0 →0+ (a − (a − a 1)) = 1 if a 6= 0. And since π0 (x) = max{[x] − x, 0} = 0 for each x, then wπ0 (0) = 0.. (d) If a 6= 0,. →0. n. er. io. al. = lim+ ((a − 1) − a) = −1. →0. Ch. sit. = lim+ (πa0 (0 + ) − πa0 (1 − )). y. →0. Nat If a = 0,. ‧. wπa (n) = lim+ (πa0 (n + ) − πa0 (n − )). engchi. i n U. v. π0 (x) = max{[x] − x, 0} = 0 ∀x. wπ0 (n) = lim+ (π00 (n + ) − π00 (n − )) = 0 − 0 = 0. →0. (e) It is clear that the support of wf on [0, 1) only possible at x = 0 or x = a, so the conclusion holds.. 4.2. R-linear Combination of Tropical 1-Periodic Functions. We first introduce some lemmas before proving the main theorem in this chapter. 25.

(187) 1 x x Lemma 4.1. Let f (x) = πa ( ), T > 0, then wf (x) = wπa ( ). T T T Proof. If. x 1 x is not a support of wπa , by Chain Rule, f 0 (x) = πa0 ( ). T T T wf (x) = lim+ (f 0 (x + ) − f 0 (x − )) →0. x+ x− 1 1 = lim+ ( πa0 ( ) − πa0 ( )) →0 T T T T 1 x x = lim+ (πa0 ( + ) − πa0 ( − )) T →0 T T T T 1 x = w πa ( ) T T. 立. 政治大. c∈(supp wf ). T. ‧. ‧ 國. 學. Lemma 4.2. A non-constant tropical periodic function has as many roots and poles in a period interval, counting multiplicities. That is, X wf (c) = 0 [0,T ). y. Nat. er. io. sit. for any tropical T -periodic function f (x). More generally, X X wf (x) = 0 wf (c) =. n. k≤c<T +k a lT[k,T +k) v i n Ch for any real constant k. engchi U c∈(supp wf ). Proof. Let f (x) be a tropical T -periodic function, {ci | i = 1, 2, · · · , k} is the support of wf (x) on [0, T ) with 0 ≤ c1 < c2 < · · · < ck < T. Hence, c0 := ck − T < c1 < c2 < · · · < ck < c1 + T := ck+1 , Since f (x) is linear in 0 − each (ci , ci+1 ), then f 0 (c+ i ) = f (ci+1 ), i = 0, 1, 2, · · · , k. In particular, 0 − 0 − 0 − f 0 (c+ k ) = f (ck+1 ) = f ((c1 + T ) ) = f (c1 ). 26.

(188) X c∈(supp wf ). wf (c) = T. k X. wf (ci ). i=1. [0,T ). =. =. k X i=1 k X. lim (f 0 (ci + ) − f 0 (ci − )). →0+. 0 − (f 0 (c+ i ) − f (ci )). i=1 0. 0 − 0 + 0 − 0 + 0 − = (f (c+ 1 ) − f (c1 )) + (f (c2 ) − f (c2 )) + · · · + (f (ck ) − f (ck )) 0 − = f 0 (c+ k ) − f (c1 ) = 0. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.2:. Geometrically, it says that the sum of all changes of the slope is zero during a period.(Figure 4.2) Theorem 4.3 (R-linear Combination Of Tropical 1-Periodic Functions). Let f (x) be an tropical 1-periodic function on R, {ci | i = 1, 2, · · · , k} is the support of wf (x) on [0, 1) with 0 ≤ c1 < c2 < · · · < ck < 1. Then k X f (x) = ( wf (ci )πci (x)) + f (0) i=1. 27.

(189) Proof. Define k X ˆ f (x) = ( wf (ci )πci (x)) + f (0) i=1. If c1 = 0, wfˆ(c1 ) = wfˆ(0) =. k X. (4.9). wf (ci )wπci (0). (4.10). i=1 k X. ) × (−1) 政治w (c大. = 0 × wf (c1 ) +. 立= (−1) × (. f. (4.11). i=2. k X. wf (ci )). (4.12). 學. i=2. ‧ 國. i. = (−1) × (−wf (c1 )) = wf (c1 ). (4.13) (4.14). ‧. k X. er. io. sit. y. Nat. (4.10) follows by (4.1), (4.11) follows by (4.4), and (4.13) follows by Lemma 4.2. If c1 6= 0, then cj 6= 0, j = 1, 2, · · · , k.. n. awl fˆ(cj ) = i=1 wf (ci)wπ (cj ) i v C h= wf (cj )wπ (cj )U n engchi ci. cj. = wf (cj ). (4.15) (4.16) (4.17). (4.15) follows by linear property of wfˆ, (4.16) and (4.17) follows by (4.3) and (4.5). And if x 6= c1 , c2 , · · · , ck , wπcj (x) = 0 for each j, hence, wfˆ(x) = 0 = wf (x). Therefore, for x ∈ [0, 1), wfˆ(x) = 0 = wf (x). And since fˆ(x) is a linear finite combination of πa , s which are of period one, so fˆ(x) is of period one too. Therefore, for all x ∈ R wfˆ(x) = 0 = wf (x). 28.

(190) By Lemma 3.2, fˆ(x) = f (x)+Ax+B for some constants A and B. Moreover, fˆ(0) = (. k X. wf (ci )πci (0)) + f (0) = 0 + f (0) = f (0).. i=1. πci (0) = 0 from (4.1), it follows B = 0. And, fˆ(1) = (. k X. wf (ci )πci (1)) + f (0) = 0 + f (0) = f (0) = f (1).. i=1. It follows A = 0. Therefore, (c )π (x)) + f (0) 政 w治大. f (x) = fˆ(x) = (. 立. k X. f. i. ci. i=1. y er. sit. wf (ci )πci (x)) + f (0). io. i=1. Nat. f (x) = (. 4 X. ‧. ‧ 國. 學. Example 4.1. Let f (x) be a tropical 1-periodic function, {ci | i = 1, 2, · · · , k} is the support of wf (x) on [0, 1) with c1 = 0.2, c2 = 0.4, c3 = 0.6, c4 = 0.8. And wf (c1 ) = 1, wf (c2 ) = −3, wf (c3 ) = 3, wf (c4 ) = −1, f (0) = 0 (Figure 4.3). Note that wf (c1 ) + wf (c2 ) + wf (c3 ) + wf (c4 ) = 1 + (−3) + 3 + (−1) = 0. Then by Theorem 4.3,. = 1 × π0.2 (x) + (−3) × π0.4 (x) + 3 × π0.6 (x) + (−1) × π0.8 (x) + 0 = π0.2 (x) − 3π0.4 (x) + 3π0.6 (x) − π0.8 (x). n. al. 4.3. Ch. engchi. i n U. v. R-linear Combination of Tropical T -Periodic Functions. Theorem 4.4 (R-linear Combination Of Tropical T -Periodic Functions). Let f (x) be an tropical T -periodic function on R, {ci | i = 1, 2, · · · , k} is the support of wf (x) on [0, T ) with 0 ≤ c1 < c2 < · · · < ck < T. Then f (x) = T × (. k X i=1. x wf (ci )π ci ( )) + f (0) T T 29.

(191) 立. 政治大 Figure 4.3:. ‧ 國. 學. k X. y. sit. io. n. al. i n U. k X wfˆ(cj ) = T × ( wf (ci )wfci (cj )). Ch. ei=1n g c h i. er. i=1. If cj 6= 0. k X x wf (ci )π ci ( )) + f (0) = T × ( wf (ci )fci (x)) + f (0) T i=1 T. Nat. fˆ(x) = T × (. ‧. x Proof. The proof is very similar to Theorem 4.3. Let fci (x) = π ci ( ), T T i = 1, 2, · · · , k. Define. v. k X 1 cj =T ×( wf (ci ) × wπ c ( )) i T T i=1 T k X cj = wf (ci )wπ c ( ) i T i=1 T cj = wf (cj )wπ c ( ) j T T = wf (cj ) × 1 = wf (cj ). 30. (4.18). (4.19). (4.20) (4.21) (4.22) (4.23).

(192) (4.19) follows by Lemma 4.1, (4.21) follows by (4.5), (4.22) follows by (4.3). If c1 = 0, wfˆ(c1 ) = wfˆ(0) =T ×( =T ×(. (4.24). k X i=1 k X. wf (ci )wfci (0)) 1 0 wπ c ( )) i T T T. wf (ci ) ×. i=1. =. 立. w (c )w 治政 c (0) 大 t. k X. f. i=1. π. wf (ci )wπ c (0) i. i=2 k X. T. (4.26). (4.27). i. wf (ci ) × (−1). ‧. ‧ 國. k X. i. 學. =0+. =. (4.25). i=2. (4.29). y. (4.30). sit. Nat. = wf (c1 ). (4.28). n. al. er. io. (4.26) follows by Lemma 4.1, (4.28) and (4.29) follows by (4.4), (4.30) follows x by Lemma 4.2. And if x 6= c1 , c2 , · · · , ck , wπ c ( ) = 0 for each j, hence, i T T wfˆ(x) = 0 = wf (x). Therefore, for all x in [0, T ), wfˆ(x) = 0 = wf (x). And x since fˆ(x) is a linear finite combination of π ci ( ) , s which are with period T T T , so is fˆ(x). Therefore, for all x ∈ R. Ch. engchi. i n U. v. wfˆ(x) = 0 = wf (x). By Lemma 3.2, fˆ(x) = f (x)+Ax+B for some constants A and B. Moreover, fˆ(0) = (. k X i=1. 0 wf (ci )π ci ( )) + f (0) = 0 + f (0) = f (0). T T. 31.

(193) π ci (0) = 0 from (4.1), it follows B = 0. And, T fˆ(T ) = (. k X i=1. T wf (ci )π ci ( )) + f (0) = 0 + f (0) = f (0). T T. It follows A = 0. Therefore, k X x ˆ f (x) = f (x) = T × ( wf (ci )π ci ( )) + f (0) T i=1 T. 立. 政治大. ‧ 國. 學. Example 4.2. Let’s consider Figure 3.2(b) now, that is, f (x) = |x|, −1 ≤ x ≤ 1; f (x + 2) = f (x) for each x ∈ R. We can find a finite R-linear combination of πa , s for it. f (x) is of period 2, and {0, 1} are the support of wf on [0, 2), wf (0) = 1 − (−1) = 2, wf (1) = −1 − 1 = −2, f (0) = 0. Therefore,. ‧. x x f (x) = 2 × (2 × π 0 ( ) + (−2) × π 1 ( )) + f (0) 2 2 2 2 x x = −4π 1 ( ) = −4π0.5 ( ) 2 2 2 1 x x 1 −x −x = −4 × max{ ([ ] − ), ([ ]−( ))} 2 2 2 2 2 2 x −x x x ] + )} = min{−2([ ] − ), −2([ 2 2 2 2 x −x = min{−2[ ] + x, −2[ ] − x} 2 2 = min{x, 2 − x} f or 0 ≤ x < 2 ( x 0≤x<1 = 2−x 1≤x<2. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. x We see that −4π0.5 ( ) is really the formula we want to find. 2 Example 4.3. Given Figure 4.4, support of wf are 0, 1, 2, 2.5, 3, 4, and wf (0) = 2 − (−2) = 4, wf (1) = 0 − 2 = −2, wf (2) = 1 − 0 = 1, 32.

(194) wf (2.5) = −1 − 1 = −2, wf (3) = 0 − (−1) = 1, wf (0) = −2 − 0) = −2, f (0) = 1. Therefore, x x x x f (x) =5 × (4 × π 0 ( ) + (−2) × π 1 ( ) + 1 × π 2 ( ) + (−2) × π 2.5 ( ) 5 5 5 5 5 5 5 5 x x + 1 × π 3 ( ) + (−2) × π 4 ( )) + f (0) 5 5 5 5 x x x x x x =20π0 ( ) − 2π0.2 ( ) + π0.4 ( ) − 2π0.5 ( ) + π0.6 ( ) − 2π0.8 ( ) + 1 5 5 5 5 5 5. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.4:. Example 4.4. Let fN (x) = (−N )

(195) (x ⊕ (−∞))

(196). N K. 1 ((x ⊕ n)

(197) 2 (x ⊕ (n − ))

(198) 2 ), 2 n=−N. 33.

(199) See Figure 4.5 for N = 1 and N = 2. By Theorem 3.3 and Theorem 4.3, lim fN (x) = 2π0 (x) + (−2)π0.5 (x) = −2π0.5 (x). N →∞. 1 1 = −2 max{ ([x] − x), ([−x] − (−x))} 2 2 = − max{[x] − x, [−x] − (−x)} = 1T (([x] − x) ⊕ ([−x] − (−x))). 政治大. 學 (b) f2 (x).. Nat. y. (a) f1 (x).. n. er. io. sit. Figure 4.5:. al. ‧. ‧ 國. 立. Ch. engchi. 34. i n U. v.

(200) Chapter 5 Application on Difference Equations 政治大立 ‧ 國. 學. In this chapter, we will consider difference equation of type y(x + 1) = y(x)

(201) c = cy(x). (c ∈ R). ‧. er. io. sit. y. Nat. It is called ultradiscrete equations in [3] and [4], and we want to find the solutions y(x) which are tropical meromorphic functions. Before it, we consider certain special tropical meromorphic functions introduced in [4], Laine and Tohge define [x]−1 X a 1 v l [x]) ), (x − + αj = α[x] (x n −i[x] + C h j=−∞ α−1 U engchi. n eα (x) := α. [x]. where α is a real number with |α| > 1. In a similar way, they also define eβ (x) :=. ∞ X j=[x]. β j −β [x] (x−[x]) =. ∞ X. β j +β [x] (1−x+[x]) = β [x] (. j=[x]+1. 1 −x+[x]), 1−β. where β is real number with |β| < 1. But now, we will multiply eα (x) by α, and then rename it by eα (x). It will see new eα (x) still have the similar properties, and it will be better because of weα (0) = α − 1, we will discover that at last.. 35.

(202) 5.1. Tropical Counterpart to The Exponential Function. Definition 5.1. Let α be a real number with |α| > 1. Define a function eα (x) on R by eα (x) := α. 1+[x]. [x] X. (x − [x]) +. αj = α1+[x] (x − [x] +. j=−∞. Then we will see Remark 5.1.. 政治大. 立. eα (m) =. α1+m α−1. ∀ m ∈ Z.. 學. ‧ 國. (a). 1 ), α−1. (5.1). (b) For x ∈ [m, m + 1), m ∈ Z,. ‧. eα (x) = α1+m x + α1+m (−m +. (5.2). sit. y. Nat. 1 ) α−1. io. al. er. (c) eα (x + 1) = αeα (x) ∀x ∈ R.. n. (d) eα (x) is continuous on R. And. Ch. m. weα (m) = α (α − 1), weα (x) = 0, x ∈ / Z.. engchi. iv n Um ∈ Z.. (5.3) (5.4). In particular, we2 (0) = 1. Proof. These follows from Definition 5.1 by a straightforward computation. 1 α1+m (a) eα (m) = α1+m (m − m + − 1) = α α−1 (b) For x ∈ [n, n + 1), n ∈ Z, 1 1 eα (x) = α1+n (x − n + ) = α1+n x + α1+n (−n + ) α−1 α−1. 36.

(203) (c) For x ∈ R, 1 ) α−1 1 ) = α1+1+[x] ((x + 1) − ([x] + 1) + α−1 1 = α × α1+[x] (x − [x] + ) α−1 = αeα (x). eα (x + 1) = α1+[x+1] ((x + 1) − [x + 1] +. (d) From (b), linearity of eα (x) on [m, m + 1) implies that it is continuous at non-integer points x, and it remains to verify that eα (x) is continuous at integer points x = m ∈ Z. Take 0 < < 1, then. 政治大 1 1. 立. α−1. eα (m − ) = αm ((m − ) − (m − 1) + It follows. ) = α1+m (. α−1. + ). 1 1 1 ) = αm (1 + − ) = α1+m ( − ) α−1 α−1 α−1 α. 學. lim+ eα (m + ) = lim+ eα (m − ) = eα (m) = →0. αm+1 α−1. y. Nat. →0. ‧. ‧ 國. eα (m + ) = α1+m (m + − m +. sit. er. io. Hence, eα (x) is continuous on R. And, by equation (5.2), for m ∈ Z, we can get e0α (m + ) = α1+m , and e0α (m − ) = αm . Therefore, weα (m) = lim+ (e0α (m + ) − e0α (m − )). n. al. →0 1+m. Ch. =α. m. −α =α. m. engchi. i (αU−n1). v. In particular, we2 (0) = α0 (2 − 1) = 1.. Example 5.1. The graph of e2 (x) is an approximate function of 21+x . See Figure 5.1, let Am be the point (m, 21+m ), m ∈ Z. e2 (x) is the graph of all line segments by joining Am and Am+1 , and e2 (x) ≥ 21+x for all x ∈ R, the eqaul sign holds if and only if x ∈ Z. The proof is given in the following. Suppose n < x = n + < n + 1, for some n ∈ Z, 0 < < 1. e2 (x) = 21+n ((n + ) − n + 1) = 21+n (1 + ) > 21+n × 2 = 21+n+ = 21+x The inequality holds because of 1 + > 2 for 0 < < 1. And if x = m ∈ Z, e2 (m) = 21+m follows from (5.1). For convenience, we omit 2 from e2 (x), and denote e(x) = e2 (x). 37.

(204) 立. 政治大. ‧. ‧ 國. 學. Figure 5.1: e2 (x) is an aproximate function of 21+x .. sit. y. Nat. Application on Difference Equations: First Order a l iv. n. er. io. 5.2. Now, let’s consider. Ch. n U i engch. y(x + 1) = y(x)

(205) c = cy(x). (c ∈ R).. (5.5). In [3], lemma 4.1, Halburd and Southall have shown that equation (5.5) admits a nonconstant tropical meromorphic soluction on R if and only if c = ±1. For convenience and completeness, we carry the proof into here and make changes slightly. Theorem 5.1. The equation (5.5) admits a nonconstant tropical meromorphic soluction on R if and only if c = ±1. Proof. If c = 0 then y ≡ 0 is the only solution. If c = 1 then y is any tropical meromorphic 1-periodic function. If u is any 2-periodic tropical meromorphic 38.

(206) function, then y(x) := u(x + 1) − u(x) is a tropical meromorphic solution of equation (5.5) with c = −1. Conversely, if h(x) is a tropical meromorphic solution function of y(x + 1) = −y(x), then there exist a 2-periodic function v(x) such that h(x) = v(x + 1) − v(x). We can define ( 0 0≤x<1 v(x) = −h(x) 1≤x<2 and v(x + 2) = v(x) for each x ∈ R. Then ( −h(x + 1) v(x + 1) = 0. 立. 0≤x<1 治政 1≤x<2 大. ( −h(x + 1) = h(x) v(x + 1) − v(x) = h(x). ‧ 國. 學. 0≤x<1 1≤x<2. ‧. Hence, by periodicity of h(x) and v(x), h(x) = v(x + 1) − v(x) for all x ∈ R. If y is nonconstant then there is an x0 ∈ R such that y 0 exists and is a nonzero integer m at x0 . It follows from equation (5.5) that for all v ∈ Z, m y 0 (x0 − v) = v . Therefore if c 6= ±1 then for sufficiently large v, 0 < c |y 0 (x0 − v)| < 1, and hence the slope is not an integer.. io. sit. y. Nat. n. al. er. Note that if we allow the solution y(x) with non-integer slope, then any extended tropical meromorphic 1-periodic function is a solution of equation y(x + 1) = y(x). And we can express the solutions with a linear combination of πa (x) ’s by Theorem 4.3. The following theorem is the case of c 6= 0, ±1.. Ch. engchi. i n U. v. Theorem 5.2. y(x + 1) = cy(x),. c 6= 0, ±1.. (5.6). Given an arbitrary extended tropical meromorphic solution f to equation (5.6) with discontinuities of slope at x1 , x2 , · · · , xk in [0, 1), then f can be represented as a linear combination of finite shifts of the function e2 (x), that is, k k X X f (x) = wf (xi )e2 (x − xi ) = wf (xi )e(x − xi ) i=1. i=1. 39.

(207) Proof. Given a non-trivial tropical meromorphic solution f to equation (5.6), there are only finitely many points x1 , x2 , · · · , xk in the interval on which wf (x) 6= 0. And at least one point x such that wf (x) 6= 0. If wf (x) = 0 for all x ∈ [0, 1), there exist > 0 such that f (x) = ax + b on [−, 1) for some constants a and b. For every δ with 0 < δ < , equation (5.6) implies y(1 − δ) = cy(−δ) a(1 − δ) + b = c(a(−δ) + b) a a(1 − δ) + b =1+ c= a(−δ) + b b − aδ. 政治大. but c is a constant, P Pak contradiction. ˆ Define f (x) = i=1 wf (xi )e2 (x − xi ) = ki=1 wf (xi )e(x − xi ), then. 立. =. k X i=1 k X. wf (xi )e(x + 1 − xi ). 學 wf (xi )ce(x − xi ). Nat. =c. wf (xi )e(x − xi ). y. i=1 k X. sit. i=1. io. er. = cfˆ(x). Hence, fˆ is a solution of equation (5.6).. n. al. Ch. engchi. When x ∈ [0, 1), we see that if x 6= xj , wfˆ(x) =. ‧. ‧ 國. fˆ(x + 1) =. k X. wf (xi )we (x − xi ) =. i=1. i n U. k X. v. wf (xi ) × 0 = 0. i=1. e(x − xi ) is the shift of e(x) to right for xi unit, we (x − xi ) 6= 0 only for x = xi on [0, 1), hence, if x 6= x1 , x2 , · · · , xk , we (x − xi ) = 0 for each i = 1, 2, · · · , k. And, for j = 1, 2, · · · , k, wfˆ(xj ) =. k X. wf (xi )we (xj − xi ). i=1. = wf (xj )we (xj − xj ) = wf (xj )we (0) = wf (xj ) × 1 = wf (xj ) 40.

(208) we (0) = 1 follows by equality (5.3). Therefore, applying Lemma 3.2 to conclude that f (x) = fˆ(x) + Ax + B for some constants A, B ∈ R. And for each x ∈ R, f (x + 1) − cf (x) = fˆ(x + 1) + A(x + 1) + B − c(fˆ(x) + Ax + B) = (fˆ(x + 1) − cfˆ(x)) + (1 − c)Ax + A + (1 − c)B = (1 − c)Ax + A + (1 − c)B = 0 We conclude A = 0 and B = 0 since c 6= 1. Hence, f = fˆ on [0, 1). If x is a real number such that n ≤ x < n + 1 for some n ∈ Z, let x = n + x0 , then 0 ≤ x0 < 1, f (x) = f (n + x0 ) = cn f (x0 ) = cn fˆ(x0 ) = fˆ(n + x0 ) = fˆ(x). Therefore, f (x) = fˆ(x) for all x ∈ R.. f (x) =. k X. 學. ‧ 國. 政治大 Remark 5.2. When applying Theorem 5.2, we prefer to write the represen立 tation for f in the form ai e(x − xi ). i=1. ‧. where ai = wf (xi ).. y y(x + 1) = y(x) + b. (b ∈ R). io. sit. Nat. Now, let’s consider the equation. (5.7). n. al. er. It is clear that y(x) = bx is a solution function of equation (5.7), and if h(x) is an extended tropical meromorphic function such that h(x+1) = h(x), then h(x) + bx is an extended tropical meromorphic solution function of equation (5.7). Moreover, all solution functions of equation (5.7) is the form h(x) + bx, where h(x) is a tropical 1-periodic function. And now the equation. Ch. engchi. i n U. v. (a, b ∈ R, a 6= 0, ±1). y(x + 1) = ay(x) + b. (5.8). can be turned into y(x + 1) − Let z(x) = y(x) −. b b = a(y(x) − ) 1−a 1−a. (5.9). b , equation (5.9) turn into 1−a z(x + 1) = az(x) 41. (5.10).

(209) By Theorem the solution function of equation (5.10) can be represented P5.2, k as z(x) = i=1 ai e(x − xi ), where 0 ≤ xi < 1, ai ∈ R \ {0}. Hence, the solution function of equation (5.8) can be represented as y(x) = (. k X. ai e(x − xi )) +. i=1. b . 1−a. Last case, consider the equation y(x + 1) = −y(x) + b. (b ∈ R).. (5.11). 政治大. b Subtract both sides by − , it turns into 2. 立. b b = −(y(x) − ) 2 2. 學. ‧ 國. y(x + 1) −. ‧. Hence, by Theorem 5.1, the solution function of equation (5.11) can be represented as b y(x) = u(x + 1) − u(x) + 2 where u(x) is any 2-periodic tropical meromorphic function. So far we have discussed all the circumstances of equation y(x + 1) = ay(x) + b. For the example of second order difference equations, one can see [4], Laine and Tohge consider equation. er. io. sit. y. Nat. n. al. for c ∈ R.. 5.3. i n y(xC + 1) + y(x − 1) = cy(x) hengchi U. v. Tropical Approximated Function. We found that each continuous function defined on R can be approximated by extended tropical meromorphic functions. Given a continuous function f (x) on R, connect the line between the point (m, f (m)) and (m + 1, f (m+1)) for each integer m. This will be a function of an extended tropical approximated of original function. The function is (f ([x + 1]) − f ([x]))(x − [x]) + f ([x]). 42. (5.12).

(210) where [x] represent the greatest integer not exceeding x. 1 If every as a line linking, then this function is n 1 1 1 f ( [n(x + )]) − f ( [nx]) 1 1 n n T fn (x) := n (x − [nx]) + f ( [nx]) 1 n n n It is continuous piecewise linear, and limn→∞ T fn (x) = f (x) for each x ∈ R. Theorem 5.3. Given a continuous function f (x) on R, define. 政治大. 1 1 1 f ( [n(x + )]) − f ( [nx]) 1 1 n n T fn (x) := n (x − [nx]) + f ( [nx]) 1 n n n. 立. ‧ 國. 學. ‧. then T fn (x) is continuous piecewise linear for each n ∈ N, and limn→∞ T fn (x) = f (x) for each x ∈ R. m Proof. If n ∈ N, x = for some m ∈ Z, then n m m T fn (x) = T fn ( ) = f ( ) = f (x). n n m m+1 If x ∈ ( , ), n n m+1 m f( ) − f( ) n n (x − m ) + f ( m ). T fn (x) = 1 n n n m−1 m If x ∈ ( , ), n n m m−1 f( ) − f( ) m−1 m−1 n n (x − ) + f( ). T fn (x) = 1 n n n. n. er. io. sit. y. Nat. al. Hence,. Ch. engchi. i n U. m lim T fn (x) = f ( ), m+ n x→( ) n 43. v.

(211) and. m m−1 m−1 m lim T fn (x) = f ( ) − f ( ) + f( ) = f ( ). m− n n n n x→( ) n Therefore, T fn (x) is continuous piecewise linear for each n ∈ N. Next, if x ∈ Z, then T fn (x) = f (x) for each n ∈ N, and clearly limn→∞ T fn (x) = f (x). If x ∈ (t, t + 1) for some t ∈ Z, continuity of f (x) implies that f (x) is uniformly continuous on [t, t + 1], given > 0, there exists δ1 > 0 such that |x − y| < δ1 and x, y ∈ [t, t + 1] implies |f (x) − f (y)| < . Let 2 1 < δ := min{δ1 , δ2 }, δ2 = min{x−t, t+1−x}, there exists N ∈ N such that N mn mn + 1 for each n ≥ N, there exists mn ∈ Z such that t < ≤x< < t+1, n n and mn mn ) + T fn ( ) − f (x)| n n mn mn ≤ |T fn (x) − T fn ( )| + |T fn ( ) − f (x)| n n mn mn + 1 ) − f( ) f( mn mn n n (x − )| + |f ( ) − f (x)| =| 1 n n n mn + 1 mn mn ≤ |f ( ) − f( )| + |f ( ) − f (x)| n n n ≤ + = 2 2. 學. ‧ 國. 立. 政治大. |T fn (x) − f (x)| = |T fn (x) − T fn (. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Therefore, limn→∞ T fn (x) = f (x) for each x ∈ R. In fact, {T fn } converges to f uniformly on any closed bounded interval. Example 5.2. If f (x) = 2x+1 , then T f1 (x) = f ([x + 1] − f ([x]))(x − [x]) + f ([x]) = (2[x+1]+1 − 2[x+1] )(x − [x]) + 2[x]+1 = 2[x]+1 (x − [x]) + 2[x]+1 = 2[x]+1 (x − [x] + 1) It is precisely the function e2 (x) of section 5.1. 44.

(212) Given first-order linear difference equations with general solutions, we can use Formula (5.12) to find their tropical approximated solutions. We can then use these tropical approximated solutions to generate all the extended tropical meromorphic solutions of the given difference equations. For the future research, we hope these results can be extended to difference equations in general.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 45. i n U. v.

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