熱帶橢圓曲線之研究 - 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. 立. 政治大. ‧. ‧ 國. 學. 熱帶橢圓曲線之研究. On Tropical Elliptic Curves n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 碩士班學生：黃明怡撰指導教授：蔡炎龍博士中華民國 104 年 1 月 9 日.

(2) 國立政治大學應用數學系黃明怡君所撰之碩士學位論文熱帶橢圓曲線之研究 On Tropical Elliptic Curves. 政治大. 立. ‧. ‧ 國. 學. 業經本委員會審議通過論文考試委員會委員：. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 指導教授：系主任：. 中華民國 104 年 1 月 9 日.

(3) 中文要在數學許多分枝中, 橢圓曲線都是一個非常重要的主題, 例如在數論及代數幾何中等等。本篇論文主要是研究熱帶幾何中的橢圓曲線。首先, 我們先討論什麼是熱帶橢圓曲線的合理定義。接著我們研究熱帶橢圓曲線上的因子理論。如同古典的情況「 , 所有」在熱帶橢圓曲線上的點和該曲線的 Picard 群是一一對應的。更進一步的說, 我們還可在熱帶橢圓曲線上給一個群的結構。最後, 我們指出幾個未來可能的研究方向。關鍵字：熱帶幾何、橢圓曲線、因子理論、Picard 群. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i n U. v.

(4) Abstract. Elliptic curves has been important studying objects in many mathematics areas,. 治政 analogue of elliptic curves. We first discuss what is大 a reasonable way to define trop立 ical elliptic curves. Then, we survey divisor theory on tropical elliptic curves. Like such as number theory and algebraic geometry. In this thesis, we study tropical. ‧ 國. 學. in classical elliptic curves, all “points” in a tropical elliptic curves are one-to-one. ‧. corresponding to the Picard group of that elliptic curves. Moreover one can define group structures on any tropical elliptic curves. Finally, we give some possible. y. Nat. sit. projects for future studies.. n. al. er. io. Keywords: tropical geometry, elliptic curve, divisor theory, Picard group. Ch. engchi. iii. i n U. v.

(5) Contents. 試委員會審定. 立. ‧ 國. iii. sit. n. al. iv. er. io List of Figures. ii. y. Nat. Contents. ‧. Abstract. i. 學. 中文要. 政治大. Ch. engchi. i n U. v. vi. 1. Introduction. 1. 2. Tropical Geometry. 2. 2.1. Tropical Semifield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 2.2. Tropical Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.3. Tropical Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. iv.

(6) 3.1. Tropical Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16. 3.2. Special Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 3.3. Tropical Picard Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. Tropical Elliptic Curves. 22. 政治大. 4.1. Classical Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2. Tropical Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3. Group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. 學. al. n. 6. y. sit. Elliptic Curve Cryptogrophy . . . . . . . . . . . . . . . . . . . . . . . . . . .. io. 5.1. 22 24 26. ‧. Classical Elliptic Curves and the Cryptogrophy. Nat. 5. 16. er. 4. Divisor Theory on Tropical Geometry. ‧ 國. 3. Ch. iv n . U . . . . .. 33 33. 5.2. Tropical Elliptic Curve Cryptogrophy . .. . . . . . . . . . . . . . .. 34. 5.3. Security Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. engchi. Conclusion. 36. Bibliography. 37. v.

(7) List of Figures. 2.1. 政治大 x = 2 and x = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 立. A tropical function f ( x ) = ( x ⊕ 2) ⊙ ( x ⊕ 3) and the “critical points” are at. 5. A tropical line with f ( x, y) = y ⊕ 4 ⊙ x ⊕ 2. . . . . . . . . . . . . . . . . . .. 8. 2.3. A Newton polytope corresponding to (0, 0), (1, 0), and (0, 1). . . . . . . . . .. 8. 2.4. As f ( x, y) = x ⊙2 ⊕ 3 ⊙ y⊙2 ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5. . . . . . . . . . .. 12. 2.5. A tropical curve with f ( x, y) = x ⊙2 ⊕ 3 ⊙ y⊙2 ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5.. ‧. ‧ 國. 學. 2.2. er. io. sit. y. Nat. iv. n. al. 13. 2.6. n U ⊙ 2 ⊙ 2 i e A tropical curve with f ( x, y) =n x g⊕ c 3h⊙ y ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5.. 14. 2.7. Smooth tropical curves of degree 2 with their dual graphs. . . . . . . . . . . .. 15. 3.1. A rational function f ( x ) =. Ch. ( x ⊕2) ⊙2 ( x ⊕1) ( x ⊕ 3 ) ⊙3. with sum of outgoing slope ord p f (if. the point has no label the ord p f = 0. . . . . . . . . . . . . . . . . . . . . . .. 19. 3.2. The tropical rational function f with sum outgoing slope at each point. . . . . .. 20. 3.3. A rational function of principal divisor on genus one tropical curve. . . . . . .. 21. 4.1. An elliptic curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. vi.

(8) 4.2. The group law in an elliptic curve. . . . . . . . . . . . . . . . . . . . . . . . .. 24. 4.3. A classical elliptic curve formula in tropical geometry. . . . . . . . . . . . . .. 26. 4.4. A tropical Elliptic curve which is smooth curve with degree 3 and genus one. .. 26. 4.5. Cut the tentacles of tropical elliptic curve and homeomorphic to a circle . . . .. 27. 4.6. Group law P ∗ Q = R on curve C with length l .. 28. 立. . . . . . . . . . . . . . . . .. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. vii. i n U. v.

(9) Chapter 1 Introduction 立. 政治大. ‧ 國. 學. A classical elliptic curve is a plane algebraic curve defined by an equation of the form. ‧. y = x3 + Ax + B,. sit. y. Nat. n. al. er. io. where A and B are in certain fields, say, rational numbers.. Ch. i n U. v. Studying elliptic curves is an important topic in many fields of mathematics. For instance, in. engchi. number theory, we are very interested in how many rational points in a given elliptic curve. Another interesting fact is that there is a group structures on the points of an elliptic curve which we will explain in this thesis. Tropical geometry is the geometry theory with the tropical semifield as the ambient field. In this thesis, we will study a version of tropical elliptic curve. First, we would like to define what a tropical elliptic curve is. Then, we will study divisor theory on these tropical curves. In particular, we would like to use tropical divisor theory to define a group law on any given tropical elliptic curve.. 1.

(10) Chapter 2 Tropical Geometry 治政大立 ‧ 國. 學. Tropical geometry is basically the geometry with the tropical semifield as the ambient field. In. ‧. this chapter, we introduce some basic notions of tropical geometry. For detail discussions, please refer to [3], [7], [11].. er. io. sit. y. Nat. al Tropical Semifield n. 2.1. Ch. engchi. i n U. v. The definition of the tropical semifield is as following. Definition 2.1.1 (Tropical Semifield) The tropical semifield is the semifield (T, ⊕, ⊙) where T = R ∪ {−∞}, and two binary operations ⊕ and ⊙ which defined by the followings.. a ⊕ b := max{ a, b}, and. a ⊙ b := a + b.. 2.

(11) Note that the addition identity in T is −∞ and the multiplication identity of T is 0. The space T n has an obvious semi-module structure over R (as a tropical semiring) with the operations of ⊕ and ⊙.. ( a1 , a2 , · · · , an ) ⊕ (b1 , b2 , · · · , bn ) := (max{ a1 , b1 }, max{ a2 , b2 }, . . . , max{ an , bn }). and. λ ⊙ ( a1 , a2 , . . . , a n ) : = ( λ + a1 , λ + a2 , . . . , λ + a n ), where λ ∈ R.. 立. 政治大. We can also define tropical project spaces just as the classical project spaces.. ‧ 國. 學. Definition 2.1.2 (Tropical Projective Space). ‧. The tropical projective n-space is defined by P n := T n+1 / ∼. For all x, y in T n+1 , the. al Tropical Polynomial n. 2.2. er. io. sit. y. Nat. equivalence relation x ∼ y holds if and only if x = λ ⊙ y, where λ ∈ R.. Ch. engchi. i n U. v. Polynomial is a basic topic in the algebraic structure, and also important. We are interesting in (the polynomial would be) what happen in tropical geometry. We will see that its graph is a combination by some segments and rays, so we would see like “linear” almost everywhere. Remark 2.2.1 Like in classical, let us see the polynomial in tropical semifield.. (a) The tropical polynomial f ( x ) in (T, ⊕, ⊙) is defided by. f (x) =. n ! i =0. λi ⊙ x ⊙i = max{λ0 , λ1 + x, λ2 + 2x, . . . , λn + nx }, 3.

(12) where λi in R for all i in {0, 1, . . . , n}. (b) In hyper tropical semifield T n , the tropical polynomial with n indeterminates x1 , x2 , . . . , xn ,. f : R n −→ R is defined by !. f (x) =. α∈A. λα ⊙ x1⊙ a1 ⊙ x2⊙ a2 ⊙ . . . ⊙ xn⊙ an. = max{λα + a1 x1 + a2 x2 + · · · + an xn | α ∈ A},. (2.2.1) (2.2.2). where A be a subset of Z n , α = ( a1 , a2 , . . . , an ) and λα in R.. 政治大 Note that for any tropical polynomials f ( x ), each term can be expressed in two ways, as in the 立 equations 2.2.1 and 2.2.2. Evaluating the value of f ( x ) is finding the maximum values of all. ‧ 國. ‧. Example 2.2.1. 學. terms.. y. Nat. Let f ( x ) = ( x ⊕ 2) ⊙ ( x ⊕ 3). Since we do not have subtraction in tropical semiring, it is. n. al. er. io. and we get. sit. reasonable to think both 2 and 3 are “roots” of f ( x ). However, by some simple calculations,. Ch. engchi. i n U. v. f ( x ) = ( x ⊕ 2) ⊙ ( x ⊕ 3). = x ⊙ ( x ⊕ 3) ⊕ 2 ⊙ ( x ⊕ 3) = x ⊙2 ⊕ 3 ⊙ x ⊕ 2 ⊙ x ⊕ 5 = x ⊙2 ⊕ 3 ⊙ x ⊕ 5 = max{2x, 3 + x, 5}. As in the Figure 2.1, we can see that 2 and 3 are exactly the points where the graph of f fails to be linear.. By above example, we observe that the points we are concerned are not linear. We give the following to define what is root in tropical polynomial. 4.

(13) 政治大. 立. ‧ 國. 學 sit. y. Nat. Definition 2.2.1. ‧. Figure 2.1: A tropical function f ( x ) = ( x ⊕ 2) ⊙ ( x ⊕ 3) and the “critical points” are at x = 2 and x = 3.. io. n. al. er. Let f ( x ) be a tropical polynomial.. f (x) =. !. α∈A. i n U. v. λα ⊙ x1⊙ a1 ⊙ x2⊙ a2 ⊙ . . . ⊙ xn⊙ an. Ch. engchi. = max{λα + a1 x1 + a2 x2 + · · · + an xn | α ∈ A},. (2.2.3). A root of a polynomial f ( x ) is when at least two terms in 2.2.3 simultaneously reach the maximum values.. First, we can see in a trivial example which has only one root Example 2.2.2 Let f ( x ) = 3 ⊙ x ⊕ 5. That is, f ( x ) = max{3 + x, 5}. It is easy to see that f ( x ) has exactly. 5.

(14) one root, namely x = 2. And we can renew the representation to be. f ( x ) = 3 ⊙ ( x ⊕ 2). = (3 ⊙ x ) ⊕ (3 ⊙ 2) = max{3 + x, 3 + 2} = max{3 + x, 5} It is an example to get the property that many tropical polynomials may express the same function. And we may write every tropical polynomial to be unique as a tropical product of tropical. 政治大. linear functions. The details of given ordering such let the representation be unique and prop-. 立. erties about root of tropical polynomial, please refer to [10].. ‧ 國. 學. Tropical Curve. ‧. 2.3. sit. y. Nat. io. al. er. In classical algebraic geometry, a variety is a set of the common roots of some given polynomials.. n. We can give a similar definition in tropical geometry. More details related to tropical varieties please refer to [3].. Ch. engchi. i n U. v. Definition 2.3.1 A tropical variety is a set of common roots of a collection of tropical polynomials.. A tropical hypersurface then is the roots of single polynomial. Definition 2.3.2 A tropical hypersurface in T n is the set of the roots of single n-variable tropical polynomial. f ( x ). We denote the hypersurface by V ( f ).. In this thesis, we focus on tropical plane curves. More precisely, a tropical curve defined on the tropical plane T2 . 6.

(15) Definition 2.3.3 (Tropical Curve) A tropical curve Γ in T2 is a tropical hypersurface in T2 . That is, it is the collection of roots of one tropical polynomial f ( x, y) of two variables. In other words, we have Γ = V ( f ).. Let us see the first example of tropical curve, which is defined by a tropical polynomial of degree one. Thus, it is reasonable to call this curve a “tropical line.” Example 2.3.1. 立. 政治大. f ( x, y) = y ⊕ 4 ⊙ x ⊕ 2. (2.3.1). = max{y, 4 + x, 2}. (2.3.2). ‧ 國. 學. We want to find for what ( x, y), the three terms y, 4 + x, and 2 attain the maximum at least. ‧. twice.. y. Nat. sit. io. First, consider the points satisfy y = 4 + x ≥ 2. Thus, we have a ray on the line y = 4 + x for. n. al. er. points ( x, y) on the line and x ≥ −2. That is a ray starting from (−2, 2) and being toward to the direction of (1, 1).. Ch. engchi. i n U. v. Next, assume y = 2 ≥ 4 + x. It is easy to see the solutions are the points ( x, y) on the line. y = 2 and x ≥ −2. Thus we get another ray starting from the same point (−2, 2) and being toward to the direction of (−1, 0). Finally, let 4 + x = 2 ≥ y. A point ( x, y) satisfying this will be points on the line x = −2 and. y ≥ 2. We get yet another ray starting from (−2, 2) and being toward to the direction (0, −1). Therefore, the tropical line we get has three rays with the direction are (1, 1), (0, −1) and. (−1, 0). The three rays intersect at exactly one point (−2, 2). The resulting picture is as shown in Figure 2.2.. 7.

(16) 政治大 Figure 2.2: A tropical line with f ( x, y) = y ⊕ 4 ⊙ x ⊕ 2. 立. ‧ 國. 學. There is an interesting “dual” relation between a tropical curve and its so called “Newton subdivision” we shall explain now. In our polynomial f ( x, y), we have exactly three terms, namely,. ‧. 2, 4 ⊙ x, and y. The exponents of ( x, y) of these terms are (0, 0), (1, 0), (0, 1), respectively.. sit. y. Nat. The convex hull of these points is the Newton polytope corresponding to the polynomial f ( x, y). Now, we draw a segment between two of these three points if they attain maximum for some. io. n. al. er. points in T2 . It is easy to see we will have exactly three line segments in our Newton polytope. Ch. i n U. v. and so we get a right triangle with vertices (0, 0), (1, 0), and (0, 1). See the Figure 2.3.. engchi. Figure 2.3: A Newton polytope corresponding to (0, 0), (1, 0), and (0, 1). 8.

(17) Consider the terms y and 4 ⊙ x in f ( x, y). We have seen that they will attain maximum at the ray starting from (−2, 2) and being toward to the direction (1, 1). The corresponding segment in our Newton polytope is the line segment link (0, 1) and (1, 0). Note that the ray in tropical curve is perpendicular to the line segment in Newton polytope. It is always the case, so the three rays in our tropical curve will be perpendicular to the corresponding line segments in the Newton polytopes. Now, we give the complete illustration for Newton Polytope. Definition 2.3.4. 政治大. 立!. f (x) =. α∈A. λα ⊙ x1⊙ a1 ⊙ x2⊙ a2 ⊙ . . . ⊙ xn⊙ an. 學. ‧ 國. Let f be a tropical polynomial, and the representation is. ‧. (See the Remark 2.2.1).. sit. al. n. Definition 2.3.5. er. io. in A.. y. Nat. The Newton polytope of f is a convex hull 1 corresponding to all exponents α = ( a1 , a2 , . . . , an ). Ch. engchi. i n U. v. The Newton subdivision of f is a subdivision of the Newton polytope of f ( x, y).. Let f ( x, y) be a tropical polynomial and let Γ = V ( f ) be the corresponding tropical curve. Each edge of the Newton subdivision of f is corresponding to one edge of the curve Γ, and they are perpendicular. Moreover, we called the tropical curve (tropical variety) is the dual graph to the subdivision. The following given example of tropical curve, which is defined by a tropical quadratic polyno1 Let. X be a set of points, and S ⊂ X . The convex hull of S is a set defined by n. n. i =1. i =1. conv(S) = { ∑ λi xi | ∑ λi = 1, λi ∈ R, xi ∈ S}.. 9.

(18) mial. Example 2.3.2. f ( x, y) = x ⊙2 ⊕ 3 ⊙ y⊙2 ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5. = max{2x, 3 + 2y, x + y, 2 + x, 3 + y, 5}. (2.3.3) (2.3.4). Similarly, we want to find for what ( x, y), the six terms 2x, 3 + 2y, x + y, 2 + x, 3 + y and 5 attain the maximum at least twice.. 立. 政治大. We take 2x and 3 + 2y to distinct whether the points ( x, y) on the line 2x = 3 + 2y attain the. ‧ 國. 學. maximum in this polynomial. That is to calculate “2x = 3 + 2y ≥ x + y”, “2x = 3 + 2y ≥. ‧. 2 + x”, “2x = 3 + 2y ≥ 3 + y”, and “2x = 3 + 2y ≥ 5”. Thus it means there is ray on the. Nat. line 2x = 3 + 2y, and x ≥ y, x ≥ 2, y ≥ 0, and x ≥ 52 ”, respectively. We have a ray starting. er. io. sit. y. at ( 52 , 1) and being toward to the direction of (1, 1).. Next, let 2x = x + y and points on such line, that is x = y, and greater than the other terms.. al. n. v i n We can list those inequalities, −C 3 ≥ y − x, x ≥ 2, x ≥ 3 and x hengchi U. ≥. 5 2.. And what can we. get? Unfortunately, we have no solution. Cause the point on the line x = y does not satisfy the inequality −3 ≥ y − x, that is −3 ! 0. If we step by step to description the process, then it is a little boring, and hard to see what happen of the event. The following table 2.1 shows all comparison. The consequent tropical variety with this function is drawn in Figure 2.6. We maybe surprise that the polynomial with degree 2 is like “tropical line”. Also, we have six terms x ⊙2 , 3 ⊙ y⊙2 ,. x ⊙ y, 2 ⊙ x, 3 ⊙ y, and 5. The exponents ( x, y) of these terms are (2, 0), (0, 2), (1, 1), (1, 0),. (0, 1), and (0, 0), respectively. The Newton polytope of these points forms a convex hull. And only three terms has maximum with their exponents are (2, 0), (0, 2), and (0, 0). Where the. 10.

(19) Table 2.1: Solution table of f ( x, y) = x ⊙2 ⊕ 3 ⊙ y⊙2 ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5 maximum terms. solutions. maximum terms. solutions. 2x = 3 + 2y. (t + 52 , t + 1), t ≥ 0. x+y = 2+x. φ. φ. x+y = 3+y. φ. 2x = 2 + x. φ. x+y = 5. φ. 2x = 3 + y. φ. 2+x = 3+y. φ. 2x = 5. ( 52 , 1 − t), t ≥ 0. 2+x = 5. φ. φ. 3+y = 5. φ. 3 + 2y = 2 + x. φ. 3 + 2y = 3 + y. φ. 3 + 2y = 5. ( 52 − t, 1), t ≥ 0. 2x = x + y. 3 + 2y = x + y. 立. 政治大. “dual” graph is shown in Figure 2.4 (b) which is really like the above example ( has degree. ‧. ‧ 國. 學. one).. The difference of “dual graph” between the two examples is that the example 2.3.2 has multiple. sit. y. Nat. length on each segment than example 2.3.1. And that means has different “weight” vector.. io. al. n. division” then we get new graph and will be interesting.. Ch. engchi. er. Refer to [9] get more completion. The Newton polytope in Figure 2.4 (a), and if it has ”full. i n U. v. We consider the general quadratic polynomial, and satisfy the following conditions. We can get the “dual graph” with “full division” that means has each triangle with 1/2 area.. f ( x, y) = λ20 ⊙ x ⊙2 ⊕ λ11 ⊙ x ⊙ y ⊕ λ02 ⊙ y⊙2 ⊕ λ10 ⊙ x ⊕ λ01 ⊙ y ⊕ λ00 , where λij ∈ R, 0 ≤ i, j ≤ 2, and. 2λ11 ≥ λ20 + λ02 , 2λ10 ≥ λ20 + λ00 , 2λ01 ≥ λ02 + λ00 . Please consulting [10] for the details. 2 In. 2. There are exactly four types “full division” in degree 2. [10], the multiple operator take the minimum, and in our thesis take the maximum. And in the segment. 11.

(20) (a) Newton polytope with six exponents corresponding to f. (b) “dual” graph of f. 政治大. Figure 2.4: As f ( x, y) = x ⊙2 ⊕ 3 ⊙ y⊙2 ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5.. 立. without degeneration. Showing the Figure 2.7.. ‧ 國. 學. In this section, we are interesting tropical curve. In Example 2.3.2, we show that solving the. ‧. equation to get variety is not simple, even get degeneration one. And we talk about “dual graph”. sit. y. Nat. can help us to resolve how many types for tropical curve. If the graph is degenerated one, then. io. n. al. er. it is not “full division” and has “weight” greater 1.. i n U. v. And we can get some conclusion about tropical curve, please refer to [7], [2] for details. Remark 2.3.1. Ch. engchi. The node of V ( f ) have three tentacles and the sum of slope of tentacle is 0.. The Newton Polytope is a convex hull of given vertices. And the Newton Polygon is from a function (tropical polynomial). Definition 2.3.6 (2, 0) − −(1, 1) − −(0, 2), if (1, 1) get the maximum with others rather than (2, 0) and (0, 2), it means λ11 + x + y ≥ λ20 + 2x λ11 + x + y ≥ λ02 + 2y and get 2λ11 ≥ λ20 + λ02 . Similarly for the other segments.. 12.

(21) 政治大 Figure 2.5: A tropical curve with f ( x, y) = x ⊕ 3 ⊙ y 立 ⊙2. ⊙2. ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5.. ‧. Definition 2.3.7. ‧ 國. triangle is 1/2.. 學. A Newton subdivision is full if the subdivision consists only triangles and the area of each. Nat. sit. n. al. er. io. Definition 2.3.8. y. We called a tropical curve Γ is smooth if the corresponding Newton subdivision is a full division.. We define the genus g of a tropical curve Γ by. Ch. engchi. i n U. v. g = | E(Γ)| − |V ( G )| + 1.. 13.

(22) 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2.6: A tropical curve with f ( x, y) = x ⊙2 ⊕ 3 ⊙ y⊙2 ⊕ x ⊙ y ⊕ 2 ⊙ x ⊕ 3 ⊙ y ⊕ 5.. 14.

(23) tropical curve of degree 2. dual graph. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2.7: Smooth tropical curves of degree 2 with their dual graphs.. 15.

(24) Chapter 3 Divisor Theory on Tropical Geometry 治政大立 ‧ 國. 學. We develop divisor theories on graphs, and since each tropical curve is naturally corresponding. sit. y. Nat. n. al. er. Tropical Divisors. io. 3.1. ‧. to a graph, so we can also define divisors on a tropical curve.. Ch. engchi. First recall the definitions and results from [4].. i n U. v. Definition 3.1.1 (Metric Graphs) A metric graph G = (V, E, l ) is a finite graph such that each edge is of positive length.. For detail discussion on tropical divisors, we refer to [1]. Definition 3.1.2 Let V (Γ) be the collection of vertices in Γ. We denote the number of elements in V (Γ) by. |V (Γ)|. Similarly, | E(Γ)| denotes the number of elements in the collection of edges E(Γ) of Γ. Definition 3.1.3 Let Γ be a tropical curve. The degree deg(v) of a point v in Γ means the number of connecting edges to the point. 16.

(25) There are at least two ways to definite tropical divisors on a tropical curve, which we call discrete tropical divisors and continuous tropical divisors. In this thesis, we mainly discuss the continuous ones, but for completeness, we give the definition of discrete tropical divisors as following. Definition 3.1.4 (Discrete Tropical Divisors) Let Γ be a tropical curve. A tropical discrete divisor D is defined by a formal linear combination of vertices of Γ with coefficients in Z. Therefore, a tropical discrete divisor D on Γ is of the form. D=. 立. 政. n. mi · vi , ∑治. i =1. 大. where vi ∈ V (Γ) and n = |V (Γ)|, and mi ∈ Z, for all i = 1, 2, . . . , n. The degree deg( D ) of. ‧ 國. deg( D ) =. n. ∑ mi .. 學. a divisors is defined by. i =1. ‧. y. sit. io. er. thesis.. Nat. Now, we define the continuous tropical divisors, which will be the divisors we consider in this. al. v i n C A continuous tropical divisor D is ahformal of finite points (not necessary i U e n glinear c hcombination n. Definition 3.1.5 (Continuous Tropical Divisors). vertices) in Γ, with the coefficients in Z. That is, a continuous tropical divisor is of the form. D=. ∑. vα ∈ A. mα · vα ,. where A is a finite subset of Γ, and mα ∈ Z, for all vα ∈ A. We denote the collection of all divisors on Γ by Div(Γ). The degree deg( D ) of a tropical divisor is defined by. deg( D ) =. ∑. mα .. vα ∈ A. From now on, when we say a divisor what we really mean is a tropical continuous divisor.. 17.

(26) 3.2. Special Divisors. Follow the classical divisor theory, we give the following definitions. Definition 3.2.1 (Effecitive Divisor) A divisor D is called an effective divisor if all coefficients of D are greater or equal to zero. We will use the notation D ≥ 0 if D is effective. Definition 3.2.2 (Canonical Divisor) Let Γ be a tropical curve. The canonical divisor K is the divisor on Γ defined by. 立. ∑. v∈Γ. ‧ 國. 學. Definition 3.2.3. 政治大 K= (deg(v) − 2) · v.. ‧. A tropical meromorphic function f on Γ is a piecewise linear function defined on Γ, and the slope of the function is an integer on all pieces.. sit. y. Nat. n. al. er. io. We refer to for [11] the reason why we give this kind of definition for meromorphic functions. Example 3.2.1. Ch. engchi. i n U. v. A tropical rational function is of course a tropical meromorphic function. As the classical situation, the function. f (x) =. ( x ⊕ 2 ) ⊙2 ( x ⊕ 1 ) ( x ⊕ 3) ⊙3. is a tropical rational function. The graph of this function is shown in figure 3.1. As we can see, the graph of this rational function is indeed piecewise linear.. Definition 3.2.4 (Order) Let f be a meromorphic function on a tropical curve Γ. The order of f at the point P in Γ is the sum of the outgoing slope at point P. We denote the order of f at P by ordP f . Definition 3.2.5 (Principal Divisor) 18.

(27) 政治大 Figure 3.1: A rational function f ( x ) = with sum of outgoing slope ord 立 point has no label the ord f = 0. ( x ⊕2) ⊙2 ( x ⊕1) ( x ⊕ 3 ) ⊙3. p. p. f (if the. ‧ 國. 學. Let Γ be a tropical curve. A divisor D in Div(Γ) is called a principle divisor if. ‧. ∑ ordP f · P.. sit. P∈Γ. y. Nat. D = (f) =. n. al. er. io. The collection of all principal divisors of Γ is denoted by Prin(Γ).. Ch. Definition 3.2.6 (Linearly Equivalent). engchi. i n U. v. Two tropical divisors D1 and D2 on a tropical curve Γ is linearly equivalent if there is a meromorphic function f on Γ such that. D1 − D2 = ( f ), which we denoted by D1 ∼ D2 . Remark 3.2.1 It is easy to check, a tropical principle divisor is of degree 0, as in the classical divisor theory. Remark 3.2.2 We give the definitions of tropical discrete principal divisor here, just for completeness. Let Γ 19.

(28) be a tropical curve. A tropical discrete meromorphic function on Γ is simply a function. f : V (Γ) −→ Z. A principle divisor is then a divisor defined by one such function. Fixed a function f , the corresponding principal divisor is. ∑. (f) =. vi ∈V ( Γ ). mi · vi ,. such that. mi =. ∑. 立. ( f (v) − f (vi )).. 政治大 v ∈ n ( vi ). ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 3.2: The tropical rational function f with sum outgoing slope at each point.. 3.3. Tropical Picard Group. We can also define the tropical Picard groups. Remark 3.3.1 Let Γ be a tropical curve. It is obvious that Div(Γ) is a free abelian group. 20.

(29) 立. 政治大. ‧ 國. 學. Figure 3.3: A rational function of principal divisor on genus one tropical curve.. ‧. Definition 3.3.1. sit. n. al. er. io. Definition 3.3.2. y. Nat. The collection of all divisors of degree k of a tropical curve Γ is denoted by Divk (Γ).. i n U. v. Let Γ be a tropical curve. The Picard Group Pic(Γ) on Γ is defined by. Ch. engchi. Pic(Γ) =. Div0 (Γ) . Prin( G ). 21.

(30) Chapter 4 Tropical Elliptic Curves 政治大立 ‧ 國. 學. In this chapter, we define the tropical elliptic curves and study the properties of them. Moreover,. ‧. we compare the theories of tropical elliptic curves with the classical ones. In particular, we shall define group laws in tropical elliptical curves.. sit. y. Nat. n. al. er. io. First, we shall review some basic facts about classical elliptic curves.. 4.1. Ch. engchi. i n U. v. Classical Elliptic Curves. An Elliptic Curve is an algebraic variety defined by a two-variable polynomial of the form:. y2 = x3 + Ax + B,. (4.1.1). where A, B are in a field F. For details please refer to Chapter 2 in [6]. An example of elliptic curves is shown in Figure 4.1. Let E be an elliptic curve over a field F. It is a smooth curve of genus one. A well-known fact is that if we add the point “∞,” which we will denote as O, then the elliptic curve E ∪ {O} 22.

(31) Figure 4.1: An elliptic curve.. 政治大. has a group structure with the identity O. From now on, when we say the elliptic E, it actually includes the point O.. 立. ‧ 國. 學. We now explain the group law of an elliptic curve. Given an elliptic curve E, for any P, Q in E. ← → We can define P + Q as followings. Suppose the line PQ intersects with the elliptic curve E at. ‧ sit. Nat. P + Q + R = O.. y. R. We define. n. al. er. io. In particular, P + Q will be the inverse of R. Now, consider a line through R such that it. i n U. v. intersects with E at exactly one point. It is easy to see that the line is through R and parallel to y. Ch. engchi. axis. If the line intersects E at R′ , then the line intersects with E at exactly three points, namely. O, R, and R′ . By our definition, we have O + R + R′ = O, and R′ is the inverse of R. Hence, we have. R′ = P + Q,. as shown in Figure 4.2. For more about classical elliptic curves, one can refer to, for example, Chapter 2 in [6]. The 23.

(32) graph is shown in Figure 4.2.. Figure 4.2: The group law in an elliptic curve.. 政治大. 立 Curves Tropical Elliptic. 學. ‧ 國. 4.2. ‧. How do we define tropical elliptic curves? First, we try to use the analogue of the equation 4.1.1. sit. io. f ( x, y) = y⊙2 ⊕ x ⊙3 ⊕ A ⊙ x ⊕ B,. n. al. er. Nat. y. to define tropical elliptic curves. That is, we consider the tropical curve defined by the equation. where A, B in R.. Ch. engchi. i n U. (4.2.1). v. Theorem 4.2.1 Let Γ be a tropical curve. Let P, Q be two points on a connected tentacle of Γ. Then P ∼ Q as divisors of Γ.. Proof. Let Γ be a tropical curve. For arbitrary tentacle L of Γ, and given any two points P, Q on the same edge of L. We shall construct a meromorphic function f such that ( f ) = P − Q, and then P ∼ Q. The points P and Q obviously divide Γ into three connected components: C1 is the component connected to P and not connected to Q; C2 is the segment PQ; and C3 is the component connected to Q but not connected to P. 24.

(33) Let f be the meromorphic function defined by. f (X) =. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨. 0,. X is in C1 ,. d( P, X ), X is in C2 , and ⎪ ⎪ ⎪ ⎪ ⎩ d( P, Q), X is in C3 ,. where d( P, X ) is a distance from P to X .. Then it is easy to see that ( f ) = P − Q. Hence, as divisors, P and Q are differed by a principle divisor, so they are linearly equivalent.. 政治大 Finally, given any two points P, Q on the same tentacle, but not on the same edge of Γ. Sup立. ‧ 國. 學. pose the tentacle has E1 , E2 , . . . , Ek edges. We know that any two points on the same edge are equivalent as divisors. Each Ei will intersect with some Ej , so it is obvious that any two points. Q are equivalent.. ‧. on Ei and Ej are equivalent as divisors. Since the tentacle is connected, so we must have P and. sit. y. Nat. n. al. er. io. By Theorem 4.2.1, it is easy to see that if a tropical curve Γ is of genus zero, then Pic(Γ) is. i n U. v. trivial. Hence, if we use the classical elliptic curve with tropical operations (equation 4.2.1), we. Ch. engchi. will end up get a tropical curve (see Figure 4.3) with trivial Picard group. That is the reason we do not use that formula to define tropical elliptic curves. Observe that a classical elliptic curve is a smooth degree 3 curve with genus one, therefore we have the following definition for tropical elliptic curves. Definition 4.2.1 (Tropical Elliptic Curve) A tropical elliptic curve is a smooth tropical curve of degree three with genus one.. An example of tropical elliptic curves is shown in Figure 4.4. We have proved any two points in the same tentacle are linearly equivalent as divisors. Therefore, we can “cut” the tentacles and get the remaining part being homeomorphic to S1 . Figure 4.5 25.

(34) (a) dual graph of f. (b) tropical curve of f. 政治大. Figure 4.3: A classical elliptic curve formula in tropical geometry.. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.4: A tropical Elliptic curve which is smooth curve with degree 3 and genus one. is an example that we cut the tentacles of a tropical elliptic curve to get a loop that is homeomorphic to S1 .. 4.3. Group law. Vigeland [2] defined a group law in the tropical elliptic curve. However, he use different definitions of tropical divisors. Pflueger [8] put a group law on tropical curves of genus one (which exactly the definition of our tropical elliptic curves). We follow Pflueger’s ideas to have the 26.

(35) Figure 4.5: Cut the tentacles of tropical elliptic curve and homeomorphic to a circle . following theorems.. 立. Definition 4.3.1. 政治大. ‧ 國. 學. Let Γ be a tropical elliptic curve, and cutting the tentacles of Γ which denoted by C. We call it. er. io. sit. Nat. Once again, the graph C is homeomorphic to unit circle S1 .. y. ‧. the core of the tropical elliptic curve Γ.. Since for any points P and Q on the same tentacle of Γ are equivalent as divisors. So for any. al. n. v i n Ch P ∈ Γ, we have a point Q in the intersection of the tentacle and C, such that P ∼ Q. Hence, we engchi U. have. Div(Γ) Div(C ) = . Prin(Γ) Prin(C ) In particular, the Picard groups of Γ and C are the same:. Pic(C ) = Pic(Γ).. We will give a group structure on C and define a isomorphism between C and Pic(C ). Since C is homeomorphic to S1 (which has an obvious group structure), so we can easily define a group law on C as followings. 27.

(36) Definition 4.3.1. Let l be the length of C. We use the group structure on R/l to introduce a group law on C as followings. Fix an arbitrary point P0 on C. For each points P, Q in C, let. x, y be the distance from P to P0 , and Q to P0 , respectively. We calculate the distance from P0 counterclockwisely. Define an operation ∗ of C by. P ∗ Q = R, where R is the point in C with the distances ( x + y) mod l from P0 , as shown in Figure 4.6.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.6: Group law P ∗ Q = R on curve C with length l . It is easy to check (C, ∗) is indeed a group, since R/l ∼ = S1 , and S1 has an obvious group structure. Theorem 4.3.1 Let C be the core of a tropical elliptic curve Γ. Any two distinct points in C are not linearly equivalent as divisors.. Proof. Suppose there are two points P, Q in C are linearly equivalent as divisors. That is, there 28.

(37) is a meromorphic function f on C, such that. ( f ) = P − Q. We let f have slope with integer number n on path from P to Q with counter clockwise direction. The distance with counter clockwise direction d( P, Q) denoted by r. Divisor P has ord P ( f ) =. 1, and then we obtain f whose slope is 1 − n on the path from Q to P (Do the counter clockwise direction. Don’t forget C is a circle!), and the length of such path is l − r. Then we can get. f ( P) − f ( Q) = nr = (1 − n)(l − r ), which implies l = nl + r, and hence r be an integer. 政治大. multiple of l . We get a contradiction. So the distinct points on C are not linearly equivalent as. 立. divisors.. ‧ 國. 學. Remark 4.3.2. Let C be the core of a tropical elliptic curve, with the length l . Fix a point P0 in C. For any. ‧. points P, Q, R in C, the distances from P0 are x, y, and z, respectively, such that x + y + z = kl. er. io. sit. y. Nat. for some positive integer k. Then P + Q + R − 3P0 is a principal divisor. Proof. Choose P0 in C, and fix it. And identify C with R/l via distance from P0 anti-clockwise. al. n. v i n to P ∈ C, and modulo l . SupposeC the sum of them is a multiple of l . Consider U h e ningR,cand i h ′ ′ ′ ′ ′ ′ ′′ x ′ , y′ , z′. x = x + tl for some t in Z, where 0 ≤ x < l . Similarly to y , z , y = y + t l , z = z + t l. for some t′ , t′′ in Z, where 0 ≤ y, z < l . Without loss of generality, 0 ≤ x ≤ y ≤ z < l , and let P, Q, R be the corresponding points on C. Let f be a piecewise continuous function on [0, l ] with f (0) = 0, and the slope 0, 1, 2, 3 for the intervals of (0, x ), ( x, y), (y, z), (z, l ). Clearly, f is well-defined. We get. f ( l ) = 0( x − 0) + 1( y − x ) + 2( z − y ) + 3( l − z ). = 3l − ( x + y + z), where 3l − ( x + y + z) is a multiple of l , denoted it by kl , where k is an integer. 29.

(38) Now, we construct a meromorphic function g on C. That is g( x ) = f ( x ) − kx and the ordP ( g) is equal to ordP ( f ) on C. The divisor on C corresponding to g is ( g) = P + Q + R − 3P0 and it is a principle divisor. Theorem 4.3.3 Let C be the core of a tropical elliptic curve, and P0 is a point in C. Let P and Q be two distinct points in C. Then P − P0 and Q − Q0 are two different elements in Pic(C ). Proof. Suppose we have two distinct points P, Q in C, such that. 治政 P−P ∼ Q−P . 大 0. 立. 0. ‧. ‧ 國. Theorem 4.3.4. 學. Therefore, P ∼ Q which is a contradiction to Theorem 4.3.1.. The group (C, ∗) is isomorphic to Pic(C ).. sit. y. Nat. n. al. er. io. Proof. We define a map ϕ : C → Pic(C ) by. Ch. i n U. ϕ( P) = P − P0 ,. engchi. v. for all P in C. Given points P, Q in C, and x, y are distance from P0 , respectively. There exists R in C by above, such that the distance from P0 to R is x + y mod l .. ϕ( P) + ϕ( Q) = ( P − P0 ) + ( Q − P0 ). (4.3.1). = ( P + Q) − 2P0. (4.3.2). = ( P + Q) − 2P0 − ( P + Q + R′ − 3P0 ) + ( R + R′ + P0 − 3P0 ). (4.3.3). = R − P0. (4.3.4). = ϕ ( P ∗ Q ).. (4.3.5) 30.

(39) In equation 4.3.3, we use Remark 4.3.2 twice, and obtain two principle divisors. P + Q + R′ − 3P0 , where the sum of distance of P, Q, R′ from P0 is l . R + R′ + P0 − 3P0 , where the sum of distance of R, R′ from P0 is l .. So ϕ is homomorphism. Given P, Q in C, ϕ( P) = ϕ( Q). Then. ϕ( P) − ϕ( Q) = P − Q. 政治= 0. 大. 立. ‧ 國. 學. We get P = Q. So ϕ is injective.. ‧. The image is Pic(C ). Clearly, Div0 (C ) is generated by P − Q for arbitrary P, Q in C. Then. sit. y. Nat. Pic(C ) is generated by the equivalent class of P − Q for arbitrary P, Q in C.. n. al. er. io. P − Q = P − Q + ( Q + Q′ + P0 − 3P0 ) − ( P + Q′ + R′ − 3P0 ) + ( R + R′ + P0 − 3P0 ). Ch. i. i n U. v. (4.3.6). e0 n− gP −c Qh′ − R′ + 3P0 + R + R′ + P0 − 3P0 (4.3.7) = P − Q + Q + Q′ + P0 − 3P = P + Q′ − P − Q′ − R′ + R + R′ − P0. (4.3.8). = R − P0. (4.3.9). In equation 4.3.6, we use Remark 4.3.2 three times, and obtain three principle divisors.. Q + Q′ + P0 − 3P0 , where the sum of distance of Q, Q′ from P0 is l . P + Q′ + R′ − 3P0 , where the sum of distance of P, Q′ , R′ from P0 is l . R + R′ + P0 − 3P0 , where the sum of distance of R, R′ from P0 is l . So, every element in Pic(C ) can be represented by P − P0 , where P is in C. Then ϕ is surjection. 31.

(40) Hence ϕ is an isomorphism.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 32. i n U. v.

(41) Chapter 5 Classical Elliptic Curves and the 治政大立 Cryptogrophy. er. io. sit. y. ‧. ‧ 國. 學. Elliptic Curve Cryptogrophy. Nat. 5.1. al. v i n C h in cryptogrophy.ULet people easy to do for one way, and other. So, we would use such function engchi n. A trapdoor function is a function means easy to compute in one way, and difficult to solve in the. send the message through the process. The cryptography on elliptic curve, which the necessary byte of key is less than RSA. Alice (as A) wants to get a message from Bob (as B). Alice choose a public elliptic curve E and point P. Also choose a random number r is an integer. The Q = rP also a public key. Now, Bob get three public key are E, P and Q. Bob choose a random integer number s. The message denoted by M. Bob calculate D = sQ + M and P′ = sP. Then sends to Alice ( D, P′ ).. 33.

(42) So, Alice get the message, and use the private key s to calculate the second one.. D − rP′ = (sQ + M ) − rP′ = srP + M − rsP = M Also let the first one to minus it. So Alice get the message M. Refer to the details in Chapter 3 of [5]. It is effective to avoid the third presence to get the message.. 治. 政 Cryptogrophy Tropical Elliptic Curve 大. 立. 學. ‧ 國. 5.2. We have descripted the group law under divisor theory in Section 4.3. Now we use that try to encrypt the message. And we public the original point O and the elliptic curve C. Take the. ‧. non-original point P , also choose an arbitrary non singular integer number r, and do the group. Nat. sit. n. al. er. io. The public key are :. y. operation get Q = rP.. Ch. O, C, P, Q. engchi. i n U. v. The private key is:. r When someone has the message M for us, take a random integer number s and calculate D =. sQ + M and P′ = sP. Then sends for us ( D, P′ ). So we get the message ( D, P′ ), and use the private key s to calculate the second one.. D − rP′ = sQ + M − rsP = srP + M − rsP = M Also let the first one to minus it. So we get the message M.. 34.

(43) 5.3. Security Issue. Unfortunately, it is too easy to get the message by the third presence. The interesting thing is when we want to check arbitrary two divisors on tropical are equivalent or not is not easy to do, and may use some way like Chip-firing Game. Now, we has such group law which is only depend on the distance from the original point on the elliptic curve. And the elliptic curve is like a circle, so just calculate the length of it.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 35. i n U. v.

(44) Chapter 6 Conclusion 立. 政治大. ‧ 國. 學. We survey and study the tropical elliptic curve theory, especially the tropical divisor theory on. ‧. tropical elliptic curves. We give a definition of tropical elliptic curves, and explain why we give such definition. Then, as in classical elliptic curves, the tropical elliptic curves also have group. y. Nat. sit. structures. Moreover, we prove that (with some modification) the points on an tropical elliptic. al. n. curve.. er. io. curve are one to one correspondence to the elements in the Picard group of the tropical elliptic. Ch. engchi. i n U. v. There is so called “elliptic curve cryptography,” so we attempted to do some kind of cryptography in tropical cases. However, our approaches failed for it is too easy to break the security. How do we use tropical elliptic curve in cryptography is one of our future projects. Since it is difficult to check if two tropical divisors are equivalent or not, so one approach might be using this property to do some kind of cryptography. Our divisor theory is different from [2], where the group law very close to the classical situations. We conjecture two different group laws are strongly related, but we have not proved that yet. We are very positive that this can be done in the near future.. 36.

(45) Bibliography [1] Omid Amini.. Reduced divisors and embeddings of tropical curves, 2010, arXiv:. 1007.5364.. 立. 政治大. [2] Magnus Dehli Vigeland. The group law on a tropical elliptic curve. Math. Scand., 104(2):. ‧ 國. 學. 188–204, 2009.. ‧. [3] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch. Math.-Verein., 108(1):3–32, 2006.. y. Nat. er. al. v i n San Ling, Huaxiong Wang, C andhChaoping Algebraic curves in cryptography. Dise n g cXing. hi U n. [5]. io. arXiv:0709.4485.. sit. [4] Jan Hladký, Daniel Král’, and Serguei Norine. Rank of divisors on tropical curves, 2010,. crete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2013.. [6] Alfred Menezes. Elliptic curve public key cryptosystems. The Kluwer International Series in Engineering and Computer Science, 234. Kluwer Academic Publishers, Boston, MA, 1993. With a foreword by Neal Koblitz, Communications and Information Theory. [7] Grigory Mikhalkin. Tropical geometry and its applications. In International Congress of Mathematicians. Vol. II, pages 827–852. Eur. Math. Soc., Zürich, 2006. [8] Nathan Pflueger. Tropical curves, 2011, preprint.. 37.

(46) [9] Jürgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer. Math. Soc., Providence, RI, 2005. [10] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009. [11] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. Taiwanese J. Math., 16(2):691–712, 2012.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 38. i n U. v.

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