行政院國家科學委員會專題研究計畫 成果報告
熱帶幾何與差分方程之研究
研究成果報告(精簡版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 99-2115-M-004-003-
執 行 期 間 ： 99 年 08 月 01 日至 100 年 09 月 30 日
執 行 單 位 ： 國立政治大學應用數學學系
計 畫 主 持 人 ： 蔡炎龍
報 告 附 件 ： 國外研究心得報告
公 開 資 訊 ： 本計畫可公開查詢
中 華 民 國 100 年 12 月 28 日
中 文 摘 要 ： 本計畫研究熱帶幾何於差分方程之應用。差分方程的解自然
是熱帶亞純函數, 因此我們討論了熱帶亞純函數的基本定義
及性質。接著我們研究解熱帶線性方程式的方法。最後我們
應用這些基礎解出一些差分方程式。
中文關鍵詞： 熱帶幾何, 熱帶亞純函數, 差分方程
英 文 摘 要 ： The project study tropical geometry objects arising
from difference equations. The solutions to
difference equations are naturally tropical
meromorphic functions. Therefore, we study the
properties of as well as techniques that can apply to
tropical meromorphic functions. As the classical
situation, solving linear systems are essential to
solving difference equations. Hence we find explicit
algorithm to solve tropical linear systems. Finally,
we solve some explicit examples of difference
equations with the technique we provide.
英文關鍵詞： tropical geometry, tropical meromorphic function,
difference equation
國科會專題研究成果報告:
熱帶幾何與差分方程之研究
計畫編號: NSC 992115M004 003
-計畫
主持人: 蔡炎龍 (政治大學應用數學系)
Chapter 1
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1.1
Introduction
The project study tropical geometry objects arising from difference equations. The solutions to difference equations are naturally tropical meromorphic func-tions. Therefore, we study the properties of as well as techniques that can apply to tropical meromorphic functions. As the classical situation, solving linear systems are essential to solving difference equations. Hence we find ex-plicit algorithm to solve tropical linear systems. Finally, we solve some exex-plicit examples of difference equations with the technique we provide.
While checking if an arbitrary Fano hypersurface K-stable or not, tropical polynomials come out naturally. It is interesting to see if we can apply our techniques to study K-stability problems.
1.2
Tropical Meromorphic Functions
The tropical semi-ring is the set R ∪ {−∞} with addition and multiplication defined as followings.
Definition 1 (tropical semiring). A tropical semiring is (T, ⊕, ), where T = R ∪ {−∞}. For any x, y ∈ T, define
• x ⊕ y := max{x, y} • x y := x + y
One can then of course define tropical polynomials: they are just classical polynomials with tropical operations. More precisely, let x = (x1, x2, . . . , xn) and i = (i1, i2, . . . , in), we denote x i1 1 x
i2
2 · · · x in n by x i. A tropical polynomial is of the form
f (x) = ⊕ X
i∈I
where I is a finite subset of (N ∪ {0})n.
A tropical hypersurface is of course the zero locus (roots) of a tropical poly-nomial. Notice that a root of a tropical polynomial is actually a point where the function fails to be linear.
Definition 2 (Tropical Hypersurface). Let f (x) be a tropical polynomial. Eval-uate this polynomial is to find the maximum of the linear forms αi+ hx, ii. A point in the tropical hypersurface Hfis the maximum of the linear forms achieve at least twice and it is exactly where the graph fails to be linear.
Motivated by [4, 5], we give the following definitions to tropical meromorphic functions.
Definition 3 (Tropical Meromorphic Function). We say that a function f is a tropical meromorphic function if
(a) f is a continuous, piecewise linear function with integer slopes on R, and (b) there exist x0∈ R such that f0(x0) is constant for all x < x0.
Definition 4 (R-Tropical Meromorphic Function). We say that a function f is a R-tropical meromorphic function if f is a continuous, piecewise linear function with integer slopes on R.
Definition 5 (Extended Tropical Meromorphic Function). We say that a func-tion f is a extended tropical meromorphic funcfunc-tion if
(a) f is a continuous, piecewise linear function on R, and (b) there exist x0∈ R such that f0(x0) is constant for all x < x0.
Definition 6 (Extended R-Tropical Meromorphic Function). We say that a function f is a extended R-tropical meromorphic function if f is a continuous, piecewise linear function on R.
While solving difference equations, one can choose one of the above defini-tions to fit the needs. We refer to our paper [8] for motivadefini-tions for this definition and also some important properties of tropical meromorphic functions.
1.3
Tropical Linear Systems
A tropical linear system is a linear system with tropical operators. Solving tropical linear systems has been discussed in [2, 3, 6]. We survey and extend these results, and be able to solve all tropical linear systems of the following forms.
T1: A x = 0 T2: A x = b
T3: A x ≤ b T4: A x = B x T5: A x ≤ B x T6: C x = D y
T7: A x ⊕ a = B x ⊕ b
The techniques of solving these linear systems are different, yet some of them are related to each other. For instance, linear systems of type T1 and T2, are basically the same. Type T4, T5, T6, T7 can be regarded as the same type of problems. Finally, type T3 is a general case of type T2 in some sense. We have all details in [9], where one can find all algorithms to solve these linear systems. For illustrating the idea, we give an example here.
Example 1. Consider the tropical linear system A x = b, where
A = 3 7 −1 −∞ 6 7 −∞ −∞ 1 0 1 −∞ , b = 15 18 13 。 Write x = x1 x2 x3 x4
. Since the entries of 4th column A(4) of A are all −∞, so x4 can be anything. For instance,
A(1) x =3 7 −1 −∞ x1 x2 x3 x4 = 15 implies that max{3 + x1, 7 + x2, −1 + x3, −∞ + x4} = 15.
Since −∞ + x4 = −∞ can not be the maximum value, so we can ignore this term. Hence the linear system we need to solve is the following:
3 7 −1 6 7 −∞ 1 0 1 x1 x2 x3 = 15 18 13 . Now, since the first row of A multiply by x is 15, we get
A(1) x1 x2 x3 = 15, which is max{3 + x1, 7 + x2, −1 + x3} = 15.
Subtracting all terms by 15, we get an equivalent expression: max{−12 + x1, −8 + x2, −16 + x3} = 0. That is, −12 −8 −16 x1 x2 x3 = 0. Do the same thing to each raw, we obtain
−12 −8 −16 −12 −11 −∞ −12 −13 −12 x1 x2 x3 = 0 0 0 .
For each column of A, find the maximum entry, and change every other entries to −∞. The solutions to the new linear system will be the same as before. The new linear system becomes:
−12 −8 −∞ −12 −∞ −∞ −12 −∞ −12 x1 x2 x3 = 0 0 0 . Finally, we can solve the linear system:
x = 12 x2 x3 x4 , where x2≤ 8 x3≤ 12 x4∈ T .
1.4
Applications to Difference Equations
Consider ultradiscrete equations
y(x + 1) = y(x) c= cy(x) (c ∈ R). (1.1) which were discussed in [4] and [5], and we want to find the solutions y(x) which are tropical meromorphic functions. To solve the equation, we consider certain special tropical meromorphic functions introduced by Laine and Tohge [5]:
eα(x) := α[x](x − [x]) + [x]−1 X j=−∞ αj= α[x](x − [x] + 1 α − 1), where α is a real number with |α| > 1. Moreover, they define eβ(x) := ∞ X j=[x] βj−β[x](x−[x]) = ∞ X j=[x]+1 βj+β[x](1−x+[x]) = β[x]( 1 1 − β−x+[x]), where β is real number with |β| < 1.
In [1], we redefine eα(x), which still has similar properties as the original one, and will be easier to use in certain cases.
Figure 1.1: e2(x) is an aproximate function of 21+x.
Example 2. Figure 1.1 shows the graph of e2(x). Now, let’s consider
Halburd and Southall [4] show that equation (1.1) admits a nonconstant tropical meromorphic soluction on R if and only if c = ±1. We gave a different proof in [1].
Theorem 1. The equation (1.1) admits a nonconstant tropical meromorphic soluction on R if and only if c = ±1.
If we allow the solutions to be an extend tropical meromorphic function, then we have the following theorem.
Theorem 2.
y(x + 1) = cy(x), c 6= 0, ±1. (1.2) Given an arbitrary extended tropical meromorphic solution f to equation (1.2) 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,
f (x) = k X i=1 wf(xi)e2(x − xi) = k X i=1 wf(xi)e(x − xi)
Now, let’s consider the equation
y(x + 1) = y(x) + b (b ∈ R) (1.3) It is clear that y(x) = bx is a solution function of equation (1.3), 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
(1.3). Moreover, all solution functions of equation (1.3) is the form h(x) + bx, where h(x) is a tropical 1-periodic function.
And now the equation
y(x + 1) = ay(x) + b (a, b ∈ R, a 6= 0, ±1) (1.4) can be turned into
y(x + 1) − b
1 − a = a(y(x) − b
1 − a) (1.5) Let z(x) = y(x) − b
1 − a, equation (1.5) turn into
z(x + 1) = az(x) (1.6) By Theorem 2, the solution function of equation (1.6) can be represented as z(x) =Pk
i=1aie(x − xi), where 0 ≤ xi < 1, ai ∈ R \ {0}. Hence, the solution function of equation (1.4) can be represented as
y(x) = ( k X i=1 aie(x − xi)) + b 1 − a. Last case, consider the equation
y(x + 1) = −y(x) + b (b ∈ R). (1.7) Subtract both sides by −b
2, it turns into y(x + 1) −b
2 = −(y(x) − b 2)
Hence, by Theorem 1, the solution function of equation (1.7) can be represented as
y(x) = u(x + 1) − u(x) + b 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 [5], Laine and Tohge consider equation
y(x + 1) + y(x − 1) = cy(x) for c ∈ R.
Chapter 2
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成果
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In this project, we study the properties of tropical meromorphic functions and published a journal paper [8]. Moreover, two graduate students wrote the meth-ods of solving linear systems and solving certain difference equations as their mater’s thesis [1, 9].
While visiting the University of California, Irvine, we found that the tech-niques we had might help to deal with K-stablility problem for Fano hypersur-faces, since while checking K-stability of Fano hypersurhypersur-faces, there naturally comes out some tropical polynomials [7]. We will continue to study the relation between K-stability and tropical geometry in the following years.
Bibliography
[1] Liang Chen. Tropical meromorphic functions and their application on dif-ference equations. Master’s thesis, National Chengchi University, 2011. [2] Guy Cohen, Jean P. Quadrat, Geert J. Olsder, and Fran¸cois Baccelli.
Syn-chronization and linearity, an algebra for discrete event systems, 1992. [3] R. A. Cuninghame-Green and P. Butkovic. The equation a ⊗ x = b ⊗ y
over (max, +). Theor. Comput. Sci., 293:3–12, February 2003.
[4] Rodney G. Halburd and Neil J. Southall. Tropical Nevanlinna theory and ultradiscrete equations. Int. Math. Res. Not. IMRN, (5):887–911, 2009. [5] Ilpo Laine and Kazuya Tohge. Tropical Nevanlinna theory and second main
theorem. Proc. Lond. Math. Soc. (3), 102(5):883–922, 2011.
[6] E. Lorenzo and M. J. de la Puente. An algorithm to describe the solution set of any tropical linear system A⊙x=B⊙xA⊙x=B⊙x. Linear Algebra and its Applications, March 2011.
[7] Zhiqin Lu. K energy and k stability on hypersurfaces. COMMUNICATIONS IN ANALYSIS AND GEOMETRY, 12:599, 2004.
[8] Yen lung Tsai. Working with one-variable tropical meromorphic functions. to appear in Taiwanese Journal of Mathematics.
[9] Jun-Bo You. On tropical linear systems. Master’s thesis, National Chengchi University, 2012.
