commit c604e458e2e11ba03f7f4d1a692eb941e0dd4d79 Author: Sidney Zhang Date: Tue Dec 23 18:21:38 2025 +0800 ctreated 20251223 diff --git a/assets/82b70efc64eb5aae18a9a0b3a9243c5a70a345b3f49f3b997ee242fe6d34e59c.png b/assets/82b70efc64eb5aae18a9a0b3a9243c5a70a345b3f49f3b997ee242fe6d34e59c.png new file mode 100644 index 0000000..ef9e7a3 Binary files /dev/null and b/assets/82b70efc64eb5aae18a9a0b3a9243c5a70a345b3f49f3b997ee242fe6d34e59c.png differ diff --git a/assets/88986ae48d1809bc035cb78d0a1b8adf7bb7c455ebebdc88a69e80e91fdd3ba5.png b/assets/88986ae48d1809bc035cb78d0a1b8adf7bb7c455ebebdc88a69e80e91fdd3ba5.png new file mode 100644 index 0000000..562d49c Binary files /dev/null and b/assets/88986ae48d1809bc035cb78d0a1b8adf7bb7c455ebebdc88a69e80e91fdd3ba5.png differ diff --git a/assets/f1cf81ff5afe4d6be1ce08d3d84861c815510b2644387e6cd70efe6be95166cf.png b/assets/f1cf81ff5afe4d6be1ce08d3d84861c815510b2644387e6cd70efe6be95166cf.png new file mode 100644 index 0000000..d4cd256 Binary files /dev/null and b/assets/f1cf81ff5afe4d6be1ce08d3d84861c815510b2644387e6cd70efe6be95166cf.png differ diff --git a/books/Introduction to Tropical Geometry _ second - Diane Maclagan_Bernd Sturmfels.pdf b/books/Introduction to Tropical Geometry _ second - Diane Maclagan_Bernd Sturmfels.pdf new file mode 100644 index 0000000..ca73663 Binary files /dev/null and b/books/Introduction to Tropical Geometry _ second - Diane Maclagan_Bernd Sturmfels.pdf differ diff --git a/books/Semirings and their Applications - Jonathan S_ Golan.pdf b/books/Semirings and their Applications - Jonathan S_ Golan.pdf new file mode 100644 index 0000000..63a508a Binary files /dev/null and b/books/Semirings and their Applications - Jonathan S_ Golan.pdf differ diff --git a/books/Tropical Mathematics.pdf b/books/Tropical Mathematics.pdf new file mode 100644 index 0000000..9c08bf9 Binary files /dev/null and b/books/Tropical Mathematics.pdf differ diff --git a/readme.md b/readme.md new file mode 100644 index 0000000..fc0b633 --- /dev/null +++ b/readme.md @@ -0,0 +1,21 @@ +# README + + + +---- + + + +这里主要是我收集的一些热带数学相关的文章和书籍。 + + + +说明： + +- 有部分文章我做了中文翻译，放在 [zh_translate](zh_translate) 这个文件夹中。 + +- 原始文件放在 [books](books) 文件夹中。 + +- [assets](assets) 文件夹放了一些md中使用的图片。 + + diff --git a/zh_translate/热带数学.md b/zh_translate/热带数学.md new file mode 100644 index 0000000..9e780b8 --- /dev/null +++ b/zh_translate/热带数学.md @@ -0,0 +1,275 @@ +# ARTICLES + +# Tropical Mathematics + +DAVID SPEYER Massachusetts Institute of Technology Cambridge, MA 02139 speyer@math.mit.edu + +BERND STURMFELS University of California at Berkeley Berkeley, CA 94720 bernd@math.berkeley.edu + +This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture was the tropical approach in mathematics. This approach was in its infancy at that time, but it has since matured and is now an integral part of geometric combinatorics and algebraic geometry. It has also expanded into mathematical physics, number theory, symplectic geometry, computational biology, and beyond. We offer an elementary introduction to this subject, touching upon arithmetic, polynomials, curves, phylogenetics, and linear spaces. Each section ends with a suggestion for further research. The proposed problems are particularly well suited for undergraduate students. The bibliography contains numerous references for further reading in this field. + +The adjective tropical was coined by French mathematicians, including Jean- Eric Pin [16], in honor of their Brazilian colleague Imre Simon [19], who was one of the pioneers in what could also be called min- plus algebra. There is no deeper meaning in the adjective tropical. It simply stands for the French view of Brazil. + +### Arithmetic + +Our basic object of study is the tropical semiring \((\mathbb{R}\cup \{\infty \} ,\oplus ,\odot)\) . As a set this is just the real numbers \(\mathbb{R}\) , together with an extra element \(\infty\) that represents infinity. However, we redefine the basic arithmetic operations of addition and multiplication of real numbers as follows: + +\[x\\oplus y\\coloneqq \\min (x,y)\\qquad \\mathrm{and}\\qquad x\\odot y\\coloneqq x + y.\] + +In words, the tropical sum of two numbers is their minimum, and the tropical product of two numbers is their sum. Here are some examples of how to do arithmetic in this strange number system. The tropical sum of 3 and 7 is 3. The tropical product of 3 and 7 equals 10. We write these as + +\[3\\oplus 7 = 3\\qquad \\mathrm{and}\\qquad 3\\odot 7 = 10.\] + +Many of the familiar axioms of arithmetic remain valid in tropical mathematics. For instance, both addition and multiplication are commutative: + +\[x\\oplus y = y\\oplus x\\qquad \\mathrm{and}\\qquad x\\odot y = y\\odot x.\] + +The distributive law holds for tropical multiplication over tropical addition:where no parentheses are needed on the right, provided we respect the usual order of operations: Tropical products must be completed before tropical sums. Here is a numerical example to illustrate: + +\[x\\odot (y\\oplus z) = x\\odot y\\oplus x\\odot z,\] + +\[3\\odot (7\\oplus 11) = 3\\odot 7 = 10,\] + +\[3\odot 7\oplus 3\odot 11 = 10\oplus 14 = 10.\] + +Both arithmetic operations have a neutral element. Infinity is the neutral element for addition and zero is the neutral element for multiplication: + +\[x\\oplus \\infty = x\\qquad \\mathrm{and}\\qquad x\\odot 0 = x.\] + +Elementary school students tend to prefer tropical arithmetic because the multiplication table is easier to memorize, and even long division becomes easy. Here are the tropical addition table and the tropical multiplication table: + +
⊕1234567 + +The three displayed identities are easily verified by noting that the following equations hold in classical arithmetic for all \(x, y \in \mathbb{R}\) : + +\[3 \\cdot \\min \\{x, y\\} = \\min \\{3x, 2x + y, x + 2y, 3y\\} = \\min \\{3x, 3y\\} .\] + +Research problem The tropical semiring generalizes to higher dimensions: The set of convex polyhedra in \(\mathbb{R}^{n}\) can be made into a semiring by taking \(\odot\) as "Minkowski sum" and \(\oplus\) as "convex hull of the union." A natural subalgebra is the set of all polyhedra that have a fixed recession cone \(C\) . If \(n = 1\) and \(C = \mathbb{R}_{\geq 0}\) , this is the tropical semiring. Develop linear algebra and algebraic geometry over these semirings, and implement efficient software for doing arithmetic with polyhedra when \(n \geq 2\) . + +## Polynomials + +Let \(x_{1}, \ldots , x_{n}\) be variables that represent elements in the tropical semiring \((\mathbb{R} \cup \{\infty \})\) , \(\oplus , \odot\) . A monomial is any product of these variables, where repetition is allowed. By commutativity and associativity, we can sort the product and write monomials in the usual notation, with the variables raised to exponents,as long as we know from context that \(x_{1}^{2}\) means \(x_{1} \odot x_{1}\) and not \(x_{1} \cdot x_{1}\) . A monomial represents a function from \(\mathbb{R}^{n}\) to \(\mathbb{R}\) . When evaluating this function in classical arithmetic, what we get is a linear function: + +\[x_{2} \\odot x_{1} \\odot x_{3} \\odot x_{1} \\odot x_{4} \\odot x_{2} \\odot x_{3} \\odot x_{2} = x_{1}^{2} x_{2}^{3} x_{3}^{2} x_{4},\] + +\[x_{2} + x_{1} + x_{3} + x_{1} + x_{4} + x_{2} + x_{3} + x_{2} = 2x_{1} + 3x_{2} + 2x_{3} + x_{4}.\] + +Although our examples used positive exponents, there is no need for such a restriction, so we allow negative integer exponents, so that every linear function with integer coefficients arises in this manner. + +FACT 1. Tropical monomials are the linear functions with integer coefficients. + +A tropical polynomial is a finite linear combination of tropical monomials: + +\[p(x_{1}, \\ldots , x_{n}) = a \\odot x_{1}^{i_{1}} x_{2}^{i_{2}} \\cdots x_{n}^{i_{n}} \\oplus b \\odot x_{1}^{j_{1}} x_{2}^{j_{2}} \\cdots x_{n}^{j_{n}} \\oplus \\cdots\] + +Here the coefficients \(a, b, \ldots\) are real numbers and the exponents \(i_{1}, j_{1}, \ldots\) are integers. Every tropical polynomial represents a function \(\mathbb{R}^{n} \to \mathbb{R}\) . When evaluating this function in classical arithmetic, what we get is the minimum of a finite collection of linear functions, namely,This function \(p: \mathbb{R}^{n} \to \mathbb{R}\) has the following three important properties: + +\[p(x_{1}, \\ldots , x_{n}) = \\min \\left(a + i_{1} x_{1} + \\cdots + i_{n} x_{n}, b + j_{1} x_{1} + \\cdots + j_{n} x_{n}, \\ldots\\right).\] + +- \(p\) is continuous,- \(p\) is piecewise-linear, where the number of pieces is finite, and- \(p\) is concave, that is, \(p(\frac{x + y}{2}) \geq \frac{1}{2} (p(\mathbf{x}) + p(\mathbf{y}))\) for all \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}\) . + +It is known that every function that satisfies these three properties can be represented as the minimum of a finite set of linear functions. We conclude: + +FACT 2. The tropical polynomials in \(n\) variables \(x_{1}, \ldots , x_{n}\) are precisely the piecewise- linear concave functions on \(\mathbb{R}^{n}\) with integer coefficients. + +As a first example consider the general cubic polynomial in one variable \(x\) ,To graph this function we draw four lines in the \((x,y)\) plane: \(y = 3x + a\) , \(y = 2x + b\) , \(y = x + c\) , and the horizontal line \(y = d\) . The value of \(p(x)\) is the smallest \(y\) - value such that \((x,y)\) is on one of these four lines, that is, the graph of \(p(x)\) is the lower envelope of the lines. All four lines actually contribute if + +\[p(x) = a\\odot x^{3}\\oplus b\\odot x^{2}\\oplus c\\odot x\\oplus d. \\quad (1)\] + +\[b - a\\leq c - b\\leq d - c. \\quad (2)\] + +These three values of \(x\) are the breakpoints where \(p(x)\) fails to be linear, and the cubic has a corresponding factorization into three linear factors: + +\[p(x) = a\\odot (x\\oplus (b - a))\\odot (x\\oplus (c - b))\\odot (x\\oplus (d - c)). \\quad (3)\] + +See FIGURE 1 for the graph and the roots of the cubic polynomial \(p(x)\) . + +![](images/88986ae48d1809bc035cb78d0a1b8adf7bb7c455ebebdc88a69e80e91fdd3ba5.png) + +Every tropical polynomial function can be written uniquely as a tropical product of tropical linear functions (in other words, the Fundamental Theorem of Algebra holds tropically). In this statement we must emphasize the word function. Distinct polynomials can represent the same function. We are not claiming that every polynomial factors as a product of linear polynomials. What we are claiming is that every polynomial can be replaced by an equivalent polynomial, representing the same function, that can be factored into linear factors. For example, the following polynomials represent the same function: + +\[x^{2}\\oplus 17\\odot x\\oplus 2 = x^{2}\\oplus 1\\odot x\\oplus 2 = (x\\oplus 1)^{2}.\] + +Unique factorization of polynomials no longer holds in two or more variables. Here the situation is more interesting. Understanding it is our next problem. + +Research problem The factorization of multivariate tropical polynomials into irreducible tropical polynomials is not unique. Here is a simple example: + +\[(0\\odot x\\oplus 0)\\odot (0\\odot y\\oplus 0)\\odot (0\\odot x\\odot y\\oplus 0)\] + +\[\qquad = (0\odot x\odot y\oplus 0\odot x\oplus 0)\odot (0\odot x\odot y\oplus 0\odot y\oplus 0).\] + +Develop an algorithm (with implementation and complexity analysis) for computing all the irreducible factorizations of a given tropical polynomial. Gao and Lauder [8] have shown the importance of tropical factorization for the problem of factoring multivariate polynomials in the classical sense. + +### Curves + +A tropical polynomial function \(p:\mathbb{R}^{n}\to \mathbb{R}\) is given as the minimum of a finite set of linear functions. We define the hypersurface \(\mathcal{H}(p)\) to be the set of all points \(\mathbf{x}\in \mathbb{R}^{n}\) at which this minimum is attained at least twice. Equivalently, a point \(\mathbf{x}\in \mathbb{R}^{n}\) lies in \(\mathcal{H}(p)\) if and only if \(p\) is not linear at \(\mathbf{x}\) . For example, if \(n = 1\) and \(p\) is the cubic in (1) with the assumption (2), then + +\[\\mathcal{H}(p) = \\big\\{b - a,c - b,d - c\\big\\} .\] + +Thus the hypersurface \(\mathcal{H}(p)\) is the set of "roots" of the polynomial \(p(x)\) . + +In this section we consider the case of a polynomial in two variables: + +\[p(x,y) = \\bigoplus_{(i,j)}c_{ij}\\odot x^{i}\\odot y^{j}.\] + +FACT 3. For a polynomial in two variables, \(p\) , the tropical curve \(\mathcal{H}(p)\) is a finite graph embedded in the plane \(\mathbb{R}^{2}\) . It has both bounded and unbounded edges, all of whose slopes are rational, and the graph satisfies a zero tension condition around each node, as follows: + +Consider any node \((x,y)\) of the graph, which we may as well take to be the origin, \((0,0)\) . Then the edges adjacent to this node lie on lines with rational slopes. On each such ray emanating from the origin consider the smallest nonzero lattice vector. Zero tension at \((x,y)\) means that the sum of these vectors is zero. + +Our first example is a line in the plane. It is defined by a polynomial: + +\[p(x,y) = a\\odot x\\oplus b\\odot y\\oplus c\\qquad \\mathrm{where} a,b,c\\in \\mathbb{R}.\] + +The curve \(\mathcal{H}(p)\) consists of all points \((x,y)\) where the function + +\[p:\\mathbb{R}^{2}\\to \\mathbb{R},\\qquad (x,y)\\mapsto \\min \\bigl (a + x,b + y,c\\bigr)\] + +is not linear. It consists of three half- rays emanating from the point \((x,y) = (c - a\)\(c - b)\) into northern, eastern, and southwestern directions. The zero tension condition amounts to \((1,0) + (0,1) + (- 1, - 1) = (0,0)\) + +Here is a general method for drawing a tropical curve \(\mathcal{H}(p)\) in the plane. Consider any term \(\gamma \odot x^{i}\odot y^{j}\) appearing in the polynomial \(p\) . We represent this term by the point \((\gamma ,i,j)\) in \(\mathbb{R}^{3}\) , and we compute the convex hull of these points in \(\mathbb{R}^{3}\) . Now project the lower envelope of that convex hull into the plane under the map \(\mathbb{R}^{3}\to \mathbb{R}^{2}\) , \((\gamma ,i,j)\mapsto (i,j)\) . The image is a planar convex polygon together with a distinguished subdivision \(\Delta\) into smaller polygons. The tropical curve \(\mathcal{H}(p)\) (actually its negative) is the dual graph to this subdivision. Recall that the dual to a planar graph is another planar graph whose vertices are the regions of the primal graph and whose edges represent adjacent regions. + +As an example we consider the general quadratic polynomial + +\[p(x,y) = a\\odot x^{2}\\oplus b\\odot xy\\oplus c\\odot y^{2}\\oplus d\\odot x\\oplus e\\odot y\\oplus f.\] + +Then \(\Delta\) is a subdivision of the triangle with vertices \((0,0)\) , \((0,2)\) , and \((2,0)\) . The lattice points \((0,1)\) , \((1,0)\) , \((1,1)\) can be used as vertices in these subdivisions. Assuming that \(a, b, c, d, e, f \in \mathbb{R}\) satisfy the conditions + +\[2b \\leq a + c, 2d \\leq a + f, 2e \\leq c + f,\] + +the subdivision \(\Delta\) consists of four triangles, three interior edges, and six boundary edges. The curve \(\mathcal{H}(p)\) has four vertices, three bounded edges, and six half- rays (two northern, two eastern, and two southwestern). In FIGURE 2, we show the negative of the quadratic curve \(\mathcal{H}(p)\) in bold with arrows. It is the dual graph to the subdivision \(\Delta\) which is shown in thin lines. + +![](images/f1cf81ff5afe4d6be1ce08d3d84861c815510b2644387e6cd70efe6be95166cf.png) + +FACT 4. Tropical curves intersect and interpolate like algebraic curves do. + +1. Two general lines meet in one point, a line and a quadric meet in two points, two quadrics meet in four points, etc. 2. Two general points lie on a unique line, five general points lie on a unique quadric, etc. + +For a general discussion of Bézout's Theorem in tropical algebraic geometry, illustrated on the MAGAZINE cover, we refer to the article [17]. + +Research problem Classify all combinatorial types of tropical curves in 3- space of degree \(d\) . Such a curve is a finite embedded graph of the form + +\[C = \\mathcal{H}(p_{1}) \\cap \\mathcal{H}(p_{2}) \\cap \\dots \\cap \\mathcal{H}(p_{r}) \\subset \\mathbb{R}^{3},\] + +where the \(p_{i}\) are tropical polynomials, \(C\) has \(d\) unbounded parallel halfrays in each of the four coordinate directions, and all other edges of \(C\) are bounded. + +## Phylogenetics + +An important problem in computational biology is to construct a phylogenetic tree from distance data involving \(n\) leaves. In the language of biologists, the labels of the leaves are called taxa. These taxa might be organisms or genes, each represented by a + +DNA sequence. For an introduction to phylogenetics we recommend books by Felsenstein [7] and Semple and Steele [18]. Here is an example, for \(n = 4\) , to illustrate how such data might arise. Consider an alignment of four genomes: + +\[\\mathrm{Human:} ACAAATGTCATTAGCGAT\\dots\] + +\[\mathrm{Mouse:} ACGTTGTCAAATAGAGAT\dots\]\[\mathrm{Rat:} ACGTAAGTCATTACACAT\dots\]\[\mathrm{Chicken:} GCACAGTCAGTAGAGCT\dots\] + +From such sequence data, computational biologists infer the distance between any two taxa. There are various algorithms for carrying out this inference. They are based on statistical models of evolution. For our discussion, we may think of the distance between any two strings as a refined version of the Hamming distance ( \(=\) the proportion of characters where they differ). In our (Human, Mouse, Rat, Chicken) example, the inferred distance matrix might be the following symmetric \(4\times 4\) - matrix: + +\[H M R C\] + +\[H 0 1.1 1.0 1.4\]\[M 1.1 0 0.3 1.3\]\[R 1.0 0.3 0 1.2\]\[C 1.4 1.3 1.2 0\] + +The problem of phylogenetics is to construct a tree with edge lengths that represent this distance matrix, provided such a tree exists. In our example, a tree does exist, as depicted in FIGURE 3, where the number next to the each edge is its length. The distance between two leaves is the sum of the lengths of the edges on the unique path between the two leaves. For instance, the distance in the tree between "Human" and "Mouse" is \(0.6 + 0.3 + 0.2 = 1.1\) , which is the corresponding entry in the \(4\times 4\) - matrix. + +![](images/82b70efc64eb5aae18a9a0b3a9243c5a70a345b3f49f3b997ee242fe6d34e59c.png) + +In general, considering \(n\) taxa, the distance between taxon \(i\) and taxon \(j\) is a positive real number \(d_{ij}\) which has been determined by some bio- statistical method. So, what we are given is a real symmetric \(n\times n\) - matrix + +\[D = \\left( \\begin{array}{cccc}0 & d_{12} & d_{13} & \\dots & d_{1n}\\ d_{12} & 0 & d_{23} & \\dots & d_{2n}\\ d_{13} & d_{23} & 0 & \\dots & d_{3n}\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\ d_{1n} & d_{2n} & d_{3n} & \\dots & 0 \\end{array} \\right).\] + +We may assume that \(D\) is a metric, meaning that the triangle inequalities \(d_{ik} \leq d_{ij} + d_{jk}\) hold for all \(i, j, k\) . This can be expressed by matrix multiplication: + +FACT 5. The matrix \(D\) represents a metric if and only if \(D \odot D = D\) . + +We say that a metric \(D\) on \(\{1, 2, \ldots , n\}\) is a tree metric if there exists a tree \(T\) with \(n\) leaves, labeled \(1, 2, \ldots , n\) , and a positive length for each edge of \(T\) , such that the distance from leaf \(i\) to leaf \(j\) is \(d_{ij}\) for all \(i, j\) . Tree metrics occur naturally in biology because they model an evolutionary process that led to the \(n\) taxa. + +Most metrics \(D\) are not tree metrics. If we are given a metric \(D\) that arises from some biological data then it is reasonable to assume that there exists a tree metric \(D_{T}\) that is close to \(D\) . Biologists use a variety of algorithms (for example, "neighbor joining") to construct such a nearby tree \(T\) from the given data \(D\) . In what follows we state a tropical characterization of tree metrics. + +Let \(X = (X_{ij})\) be a symmetric matrix with zeros on the diagonal whose \(\binom{n}{2}\) distinct off- diagonal entries are unknowns. For each quadruple \(\{i, j, k, l\} \subset \{1, 2, \ldots , n\}\) we consider the following tropical polynomial of degree two: + +\[p_{ijkl} = X_{ij} \\odot X_{kl} \\oplus X_{ik} \\odot X_{jl} \\oplus X_{il} \\odot X_{jk}. \\quad (4)\] + +This polynomial is the tropical Grassmann- Plücker relation, and it is simply the tropical version of the classical Grassmann- Plücker relation among the \(2 \times 2\) - subdeterminants of a \(2 \times 4\) - matrix [14, Theorem 3.20]. + +It defines a hypersurface \(\mathcal{H}(p_{ijkl})\) in the space \(\mathbb{R}^{\binom{n}{2}}\) . The tropical Grassmannian is the intersection of these \(\binom{n}{4}\) hypersurfaces. It is denoted + +\[G_{r2,n} = \\bigcap_{1 \\leq i < j < k < l \\leq n} \\mathcal{H}(p_{ijkl}).\] + +This subset of \(\mathbb{R}^{\binom{n}{2}}\) has the structure of a polyhedral fan, which means that it is the union of finitely many convex polyhedral cones that fit together nicely. + +FACT 6. A metric \(D\) on \(\{1, 2, \ldots , n\}\) is a tree metric if and only if its negative \(X = - D\) is a point in the tropical Grassmannian \(G_{r2,n}\) . + +The statement is a reformulation of the Four Point Condition in phylogenetics, which states that \(D\) is a tree metric if and only if, for all \\(1 \\leq i < j < k < l \\leq n\\) , the maximum of the three numbers \(D_{ij} + D_{kl}\) , \(D_{ik} + D_{jl}\) , and \(D_{il} + D_{jk}\) is attained at least twice. For \(X = - D\) , this means that the minimum of the three numbers \(X_{ij} + X_{kl}\) , \(X_{ik} + X_{jl}\) , and \(X_{il} + X_{jk}\) is attained at least twice, or, equivalently, \(X \in \mathcal{H}(p_{ijkl})\) . The tropical Grassmannian \(G_{r2,n}\) is also known as the space of phylogenetic trees [3, 14, 20]. The combinatorial structure of this beautiful space is well studied and well understood. + +Often, instead of measuring the pairwise distances between the various taxa, it can be statistically more accurate to consider all \(r\) - tuples of taxa and jointly measure the dissimilarity within each \(r\) - tuple. For example, in the above tree, the joint dissimilarity of the triple {Human, Mouse, Rat} is 1.2, the sum of the lengths of all edges in the subtree containing the mouse, human, and rat. Lior Pachter and the first author showed in [15] that it is possible to reconstruct the tree from the data for all \(r\) - tuples, as long as \(n \geq 2r - 1\) . + +At this point in the original 2004 lecture notes, we had posed a research problem: to characterize the image of the given embedding of \(G_{r2,n}\) into \(\mathbb{R}^{\binom{n}{r}}\) , particularly in the case \(r = 3\) . Since then, Christiano Bocci and Filip Cools [4] have solved the problem for \(r = 3\) , and they made significant progress on the problem for higher \(r\) . While there + +is still work to be done, we now suggest the following less studied problem, borrowed from the end of [14, Chapter 3]. + +Research problem We say that a metric \(D\) has phylogenetic rank \(\leq k\) if there exist \(k\) tree metrics \(D^{(1)}\) , \(D^{(2)}\) , ..., \(D^{(k)}\) such that + +\[D_{i j} = \\max \\bigl (D_{i j}^{(1)},D_{i j}^{(2)},\\ldots ,D_{i j}^{(k)}\\bigr)\\qquad \\mathrm{for~all~}1\\leq i,j\\leq n.\] + +Equivalently, the matrix \(X = - D\) is the sum of the matrices \(X^{(i)} = - D^{(i)}\) : + +\[X = X^{(1)}\\oplus X^{(2)}\\oplus \\dots \\oplus X^{(k)}.\] + +The aim of the notion of phylogenetic rank is to model distance data that is a mixture of \(k\) different evolutionary histories. The set of metrics of phylogenetic rank \(\leq k\) is a polyhedral fan in \(\mathbb{R}^{(2)}\) . Compute this fan, and explore its combinatorial, geometric, and topological properties, especially for \(k = 2\) . + +## Tropical linear spaces + +Generalizing our notion of a line, we define a tropical hyperplane to be a subset of \(\mathbb{R}^{n}\) of the form \(\mathcal{H}(\ell)\) , where \(\ell\) is a tropical linear function in \(n\) unknowns: + +\[\\ell (x) = a_{1}\\odot x_{1}\\oplus a_{2}\\odot x_{2}\\oplus \\dots \\oplus a_{n}\\odot x_{n}.\] + +Here \(a_{1},\ldots ,a_{n}\) are arbitrary real constants. Solving linear equations in tropical mathematics means computing the intersection of finitely many hyperplanes \(\mathcal{H}(\ell)\) . It is tempting to define tropical linear spaces simply as intersections of tropical hyperplanes. However, this would not be a good definition because such arbitrary intersections can have mixed dimension, and they do not behave the way linear spaces do in classical geometry. + +A better notion of tropical linear space is derived by allowing only those intersections of hyperplanes that are "sufficiently complete," in a sense we explain later. The definition we offer directly generalizes our discussion about phylogenetics. The idea is that phylogenetic trees are lines in tropical projective space, whose Plücker coordinates \(X_{i j}\) are the negated pairwise distances \(d_{i j}\) . + +We consider the \(\binom{n}{d}\) - dimensional space \(\mathbb{R}^{\binom{n}{d}}\) whose coordinates \(X_{i_{1}\dots i_{d}}\) are indexed by \(d\) - element subsets \(\{i_{1},\ldots ,i_{d}\}\) of \(\{1,2,\ldots ,n\}\) . Let \(S\) be any \((d - 2)\) - element subset of \(\{1,2,\ldots ,n\}\) and let \(i,j,k\) , and \(l\) be any four distinct indices in \(\{1,\ldots ,n\} \backslash S\) . The corresponding three- term Grassmann Plücker relation \(p_{S,i j k l}\) is the following tropical polynomial of degree two: + +\[p_{S,i j k l} = X_{S i j}\\odot X_{S k l}\\oplus X_{S i k}\\odot X_{S j l}\\oplus X_{S i l}\\odot X_{S j k}. \\quad (5)\] + +We define the Dressian to be the intersection + +\[D_{r d,n} = \\bigcap_{S,i,j,k,l}\\mathcal{H}(p_{S,i j k l})\\subset \\mathbb{R}^{\\binom{n}{d}},\] + +where the intersection is over all \(S\) , \(i\) , \(j\) , \(k\) , \(l\) as above. The term Dressian refers to Andreas Dress, an algebraist who now works in computational biology. For relevant references to his work and further details see [11]. + +Note that in the special case \(d = 2\) we have \(S = \emptyset\) , the polynomial (5) is the four point condition in (4). In this special case, \(D r_{2,n} = G r_{2,n}\) , and this is precisely the space of phylogenetic trees discussed previously. + +We now fix an arbitrary point \(X\) with coordinates \((X_{i_{1}\dots i_{d}})\) in the Dressian \(D_{r_{d,n}}\) . For any \((d + 1)\) - subset \(\{j_{0}, j_{1}, \ldots , j_{d}\}\) of \(\{1, 2, \ldots , n\}\) we consider the following tropical linear form in the variables \(x_{1}, \ldots , x_{n}\) : + +\[\\ell_{j_{0}j_{1}\\dots j_{d}}^{X} = \\bigoplus_{r = 0}^{d}X_{j_{0}\\dots \\widehat{j_{r}}\\dots j_{d}}\\odot x_{r}, \\quad (6)\] + +where the \(\widehat{\cdot}\) means to omit \(j_{r}\) . The tropical linear space associated with the point \(X\) is the following set: + +\[L_{X} = \\bigcap \\mathcal{H}(\\ell_{j_{0}j_{1}\\dots j_{n}}^{X})\\subset \\mathbb{R}^{n}.\] + +Here the intersection is over all \((d + 1)\) - subsets \(\{j_{0}, j_{1}, \ldots , j_{d}\}\) of \(\{1, 2, \ldots , n\}\) . + +The tropical linear spaces are precisely the sets \(L_{X}\) where \(X\) is any point in \(D_{r_{d,n}} \subset \mathbb{R}_{d}^{(n)}\) . These objects are studied in detail in [21] and [11]. The "sufficient completeness" referred to in the first paragraph of this section means that we need to solve linear equations using the above formula for \(L_{X}\) , in order for an intersection of hyperplanes actually to be a linear space. The definition of linear space given here is more inclusive than the one used elsewhere [6, 17, 20], where \(L_{X}\) was required to come from ordinary algebraic geometry over a field with a suitable valuation. + +For example, a 3- dimensional tropical linear subspace of \(\mathbb{R}^{n}\) (a.k.a. a two- dimensional plane in tropical projective \((n - 1)\) - space) is the intersection of \(\binom{n}{4}\) tropical hyperplanes, each of whose defining linear forms has four terms: + +\[\\ell_{j_{0}j_{1}j_{2}j_{3}}^{X} = X_{j_{0}j_{1}j_{2}}\\odot x_{j_{3}}\\oplus X_{j_{0}j_{1}j_{3}}\\odot x_{j_{2}}\\oplus X_{j_{0}j_{2}j_{3}}\\odot x_{j_{1}}\\oplus X_{j_{1}j_{2}j_{3}}\\odot x_{j_{0}}.\] + +We note that even the very special case when each coordinate of \(X\) is either 0 (the multiplicative unit) or \(\infty\) (the additive unit) is really interesting. Here \(L_{X}\) is a polyhedral fan known as the Bergman fan of a matroid [1]. + +Tropical linear spaces have many of the properties of ordinary linear spaces. First, they are pure polyhedral complexes of the correct dimension: + +FACT 7. Each maximal cell of the tropical linear space \(L_{X}\) is \(d\) - dimensional. + +Every tropical linear space \(L_{X}\) determines its vector of tropical Plucker coordinates \(X\) uniquely up to tropical multiplication (= classical addition) by a common scalar. If \(L\) and \(L^{\prime}\) are tropical linear spaces of dimensions \(d\) and \(d^{\prime}\) with \(d + d^{\prime}\geq n\) , then \(L\) and \(L^{\prime}\) meet. It is not quite true that two tropical linear spaces intersect in a tropical linear space but it is almost true. If \(L\) and \(L^{\prime}\) are tropical linear spaces of dimensions \(d\) and \(d^{\prime}\) with \(d + d^{\prime}\geq n\) and \(v\) is a generic small vector then \(L\cap (L^{\prime} + v)\) is a tropical linear space of dimension \(d + d^{\prime} - n\) . Following [17], it makes sense to define the stable intersection of \(L\) and \(L^{\prime}\) by taking the limit of \(L\cap (L^{\prime} + v)\) as \(v\) goes to zero, and this limit will again be a tropical linear space of dimension \(d + d^{\prime} - n\) . + +It is not true that a \(d\) - dimensional tropical linear space can always be written as the intersection of \(n - d\) tropical hyperplanes. The definition shows that \(\binom{n}{d+1}\) hyperplanes are always enough. At this point in the original 2004 lecture notes, we had asked: What is the minimum number of tropical hyperplanes needed to cut out any tropical linear space of dimension \(d\) in \(n\) - space? Are \(n\) hyperplanes always enough? These questions were answered by Tristram Bogart in [2, Theorem 2.10], and a more refined combinatorial analysis was given by Josephine Yu and Debbie Yuster in [22]. Instead of posing a new research problem, we end this article with a question. + +Are there any textbooks on tropical geometry? As of June 2009, there seem to be no introductory texts on tropical geometry, despite the elementary nature of the basic + +definitions. The only book published so far on tropical algebraic geometry is the volume [10] which is based on an Oberwolfach seminar held in 2004 by Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin. That book emphasizes connections to topology and real algebraic geometry. Several expository articles offer different points of entry. In addition to [17], we especially recommend the expositions by Andreas Gathmann [9] and Eric Katz [12]. These are aimed at readers who have a background in algebraic geometry. Grigory Mikhalkin is currently writing a research monograph on tropical geometry for the book series of the Clay Mathematical Institute, while Diane Maclagan and the second author have begun a book project titled Introduction to Tropical Geometry. Preliminary manuscripts can be downloaded from the authors' homepages. In fall 2009, the Mathematical Sciences Research Institute (MSRI) in Berkeley will hold a special semester on Tropical Geometry. + +Acknowledgment. Speyer was supported in this work by a Clay Mathematics Institute Research Fellowship, Sturmfels by the National Science Foundation (DMS- 0456960, DMS- 0757236). + +### REFERENCES + +1. F. Ardila and C. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combinatorial Theory Ser. B 96 (2006) 38-49. 2. T. Bogart, A. Jensen, D. Speyer, B. Sturmfels, and R. Thomas, Computing tropical varieties, J. Symbolic Computation 42 (2007) 54-73. 3. L. Billera, S. Holmes, and K. Vogtman, Geometry of the space of phylogenetic trees, Advances in Applied Math. 27 (2001) 733-767. 4. C. Bocci and F. Cools, A tropical interpretation of \(m\)-dissimilarity maps, preprint, arXiv:0803.2184. 5. P. Butković, Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl. 367 (2003) 313-335. 6. M. Develin, F. Santos, and B. Sturmfels, On the rank of a tropical matrix, pages 213-242 in Combinatorial and Computational Geometry, Mathematical Sciences Research Institute Publication, Vol. 52, Cambridge Univ. Press, Cambridge, 2005. 7. J. Felsenstein, Inferring Phylogenies, Sinauer Associates, Sunderland, MA, 2003. 8. S. Gao and A. Lauder, Decomposition of polytopes and polynomials, Discrete and Computational Geometry 26 (2001) 89-104. 9. A. Gathmann, Tropical algebraic geometry, Jahresbericht der Deutschen Mathematiker-Vereinigung 108 (2006) 3-32. 10. I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical Algebraic Geometry, Oberwolfach Seminars Series, Vol. 35, Birkhäuser, Basel, 2007. 11. S. Hermann, A. Jensen, M. Joswig, and B. Sturmfels, How to draw tropical planes, preprint, arxiv:0808.2383. 12. E. Katz, A tropical toolkit, to appear in Expositiones Mathematicae, arXiv:math/0610878. 13. G. Mikhalkin, Enumerative tropical geometry in \(\mathbb{R}^2\) , J. Amer. Math. Soc. 18 (2005) 313-377. 14. L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge Univ. Press, Cambridge, 2005. 15. L. Pachter and D. Speyer, Reconstructing trees from subtree weights, Appl. Math. Lett. 17 (2004) 615-621. 16. J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50-69, Publ. Newton Inst., Vol. 11, Cambridge Univ. Press, Cambridge, 1998. 17. J. Richter-Gebert, B. Sturmfels, and T. Theobald, First steps in tropical geometry, pages 289-317 in Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, Vol. 377, American Mathematical Society, Providence, RI, 2005. 18. C. Semple and M. Steel, Phylogenetics, Oxford University Press, Oxford, 2003. 19. I. Simon, Recognizable sets with multiplicities in the tropical semiring, pages 107-120 in Mathematical Foundations of Computer Science (Carlsbad, 1988), Lecture Notes in Computer Science, Vol. 324, Springer, Berlin, 1988. 20. D. Speyer and B. Sturmfels, The tropical Grassmannian, Advances in Geometry 4 (2004) 389-411. 21. D. Speyer, Tropical linear spaces, SIAM J. Disc. Math. 22 (2008) 1527-1558. 22. J. Yu and D. Yuster, Representing tropical linear spaces by circuits, in Formal Power Series and Algebraic Combinatorics (FPSAC '07), Proceedings, Tianjin, China, 2007. \ No newline at end of file diff --git a/zh_translate/热带数学_translated.md b/zh_translate/热带数学_translated.md new file mode 100644 index 0000000..84262c5 --- /dev/null +++ b/zh_translate/热带数学_translated.md @@ -0,0 +1,297 @@ +# 文章 + +# 热带数学 + +大卫·斯皮尔 麻省理工学院 剑桥，马萨诸塞州 02139 speyer@math.mit.edu + +伯恩德·斯特姆费尔斯 加州大学伯克利分校 伯克利，加利福尼亚州 94720 bernd@math.berkeley.edu + +本文基于伯恩德·斯特姆费尔斯于2004年7月22日在犹他州帕克城所作的克莱数学高级学者讲座。该讲座的主题是数学中的热带方法。彼时，这一方法尚处萌芽阶段，但此后已发展成熟，如今已成为几何组合学与代数几何中不可或缺的一部分。其应用更延伸至数学物理、数论、辛几何、计算生物学等诸多领域。我们将对这一学科作基础性介绍，内容涵盖算术、多项式、曲线、系统发育学及线性空间。每一章节末尾均附有进一步研究的建议，所提问题尤其适合本科生探讨。文末参考文献亦列出大量该领域的延伸阅读资料。 + +“热带”这一形容词由法国数学家首创，其中包括让-埃里克·品[16]，旨在纪念他们的巴西同事伊姆雷·西蒙[19]——西蒙可谓“极小-加法代数”领域的先驱之一。“热带”一词本身并无深奥含义，它仅仅代表了法国人对巴西的浪漫印象。 + +### 算术 + +我们研究的基本对象是热带半环 $(\mathbb{R}\cup \{\infty \} ,\oplus ,\odot)$。作为集合，它由实数集 $\mathbb{R}$ 外加一个代表无穷大的元素 $\infty$ 构成。然而，我们重新定义了实数加法与乘法这两种基本算术运算，具体如下： + +$x\oplus y\coloneqq \min (x,y)\qquad \text{和}\qquad x\odot y\coloneqq x + y$。 + +换言之，两数的热带和取其最小值，而两数的热带积则取其和。以下举例说明如何在这一奇特的数系中进行运算：3与7的热带和为3，热带积则为10。我们将其记作： + +$3\oplus 7 = 3\qquad \text{和}\qquad 3\odot 7 = 10.$ + +许多熟悉的算术公理在热带数学中依然成立。例如，加法与乘法均满足交换律： + +$x\oplus y = y\oplus x\qquad \text{和}\qquad x\odot y = y\odot x.$ + +分配律亦适用于热带乘法对热带加法的运算：只要遵循通常的运算顺序——先完成热带积运算，再进行热带和运算——则右侧无需括号。以下用数值示例说明： + +$x\odot (y\oplus z) = x\odot y\oplus x\odot z,$ + +$3\odot (7\oplus 11) = 3\odot 7 = 10,$ + +$3\odot 7\oplus 3\odot 11 = 10\oplus 14 = 10.$ + +两种算术运算均存在单位元。无穷大是加法的单位元，零则是乘法的单位元： + +$x\oplus \infty = x\qquad \text{和}\qquad x\odot 0 = x.$ + +小学生往往更偏爱热带算术，因为其乘法表更易记忆，甚至连长除法也变得简单。以下是热带加法表与热带乘法表： + +| $\oplus$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | +| -------- | --- | --- | --- | --- | --- | --- | --- | +| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | +| 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | +| 3 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | +| 4 | 1 | 2 | 3 | 4 | 4 | 4 | 4 | +| 5 | 1 | 2 | 3 | 4 | 5 | 5 | 5 | +| 6 | 1 | 2 | 3 | 4 | 5 | 6 | 6 | +| 7 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | + +| $\odot$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | +| ------- | --- | --- | --- | --- | --- | --- | --- | +| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | +| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +| 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | +| 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | +| 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | +| 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | +| 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | + +注意到对于所有 $x, y \in \mathbb{R}$，下列等式在经典算术中均成立，由此可轻松验证上述三个恒等式： + +$3 \cdot \min \{x, y\} = \min \{3x, 2x + y, x + 2y, 3y\} = \min \{3x, 3y\} .$ + +研究问题 热带半环可推广至高维情形：$(\mathbb{R}^{n}$ 中的凸多面体集合，若取 $\odot$ 为“闵可夫斯基和”、$\oplus$ 为“并集的凸包”，便可构成一个半环。其中，所有具有固定回收锥 $C$ 的多面体构成一个自然的子代数。当 $n = 1$ 且 $C=\mathbb{R}_{\geq 0}$ 时，此结构即为热带半环。我们的目标是在这类半环上建立线性代数与代数几何理论，并开发高效算法软件，以处理 $n \geq 2$ 时多面体的算术运算。 + +## 多项式 + +令 $x_{1}, \ldots, x_{n}$ 为表示热带半环 $(\mathbb{R} \cup \{\infty\}, \oplus, \odot)$ 中元素的变量。单项式是这些变量的任意乘积（允许重复）。借助交换律与结合律，我们可对乘积进行排序，并沿用通常的指数记法书写单项式——仅需注意，在上下文中 $x_{1}^{2}$ 意指 $x_{1} \odot x_{1}$，而非通常的乘法 $x_{1} \cdot x_{1}$。每个单项式对应一个从 $\mathbb{R}^{n}$ 到 $\mathbb{R}$ 的函数。若以经典算术方式求值，该函数实为线性函数： + +$x_{2} \odot x_{1} \odot x_{3} \odot x_{1} \odot x_{4} \odot x_{2} \odot x_{3} \odot x_{2} = x_{1}^{2} x_{2}^{3} x_{3}^{2} x_{4},$ + +$x_{2} + x_{1} + x_{3} + x_{1} + x_{4} + x_{2} + x_{3} + x_{2} = 2x_{1} + 3x_{2} + 2x_{3} + x_{4}.$ + +尽管之前的例子都采用了正指数，但这一限制并非必需，因此我们允许指数为负整数，从而使得所有具有整数系数的线性函数都能以此形式表示。 + +事实1. 热带单项式即指具有整数系数的线性函数。 + +热带多项式是热带单项式的有限线性组合： + +$p(x_{1}, \ldots , x_{n}) = a \odot x_{1}^{i_{1}} x_{2}^{i_{2}} \cdots x_{n}^{i_{n}} \oplus b \odot x_{1}^{j_{1}} x_{2}^{j_{2}} \cdots x_{n}^{j_{n}} \oplus \cdots$ + +其中系数 $a, b, \ldots$ 为实数，指数 $i_{1}, j_{1}, \ldots$ 为整数。每个热带多项式对应一个函数 $\mathbb{R}^{n} \to \mathbb{R}$。若用经典算术方式求值，所得结果实为一组有限线性函数的最小值，亦即该函数 + +$p(x_{1}, \ldots , x_{n}) = \min \left(a + i_{1} x_{1} + \cdots + i_{n} x_{n},\; b + j_{1} x_{1} + \cdots + j_{n} x_{n},\; \ldots\right).$ + +$p: \mathbb{R}^{n} \to \mathbb{R}$ 具有以下三个重要性质： + +- $p$ 是连续的； +- $p$ 是分段线性的，且分段数目有限； +- $p$ 是凹函数，即对所有 $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$，满足 $p\left(\frac{\mathbf{x} + \mathbf{y}}{2}\right) \geq \frac{1}{2} \bigl(p(\mathbf{x}) + p(\mathbf{y})\bigr)$。 + +已知任何满足这三条性质的函数，皆可表示为有限个线性函数的最小值。由此可得结论： + +事实 2. 在 $n$ 个变量 $x_{1}, \ldots, x_{n}$ 上的热带多项式，恰好就是定义在 $\mathbb{R}^{n}$ 上、具有整数系数的分段线性凹函数。 + +作为第一个例子，考虑单变量 $x$ 的一般三次多项式。为了绘制这个函数的图像，我们在 $(x, y)$ 平面上画出四条直线：$y = 3x + a$，$y = 2x + b$，$y = x + c$，以及水平线 $y = d$。$p(x)$ 的值是使得点 $(x, y)$ 位于这四条直线之一上的最小 $y$ 值；也就是说，$p(x)$ 的图像就是这些直线的下包络。如果满足条件 + +$p(x) = a \odot x^{3} \oplus b \odot x^{2} \oplus c \odot x \oplus d, \quad (1)$ + +$b - a \leq c - b \leq d - c, \quad (2)$ + +那么所有四条直线实际上都会对图像有所贡献。这三个 $x$ 的值是 $p(x)$ 不再保持线性的断点，并且该三次多项式可以相应地分解为三个线性因子： + +$p(x) = a \odot (x \oplus (b - a)) \odot (x \oplus (c - b)) \odot (x \oplus (d - c)). \quad (3)$ + +三次多项式 $p(x)$ 的图像及其根见图1。 + +![](images/88986ae48d1809bc035cb78d0a1b8adf7bb7c455ebebdc88a69e80e91fdd3ba5.png) + +每个热带多项式函数都可以唯一地表示为热带线性函数的热带乘积（换言之，代数基本定理在热带意义下依然成立）。在这一陈述中，我们必须强调“函数”这个词。不同的多项式可以表示同一个函数。我们并非声称每个多项式都能分解为线性多项式的乘积，而是主张：每个多项式都可以被一个等价的、表示同一函数的多项式所替代，并且该等价多项式可以分解为线性因子。例如，以下多项式表示同一个函数： + +$x^{2} \oplus 17 \odot x \oplus 2 = x^{2} \oplus 1 \odot x \oplus 2 = (x \oplus 1)^{2}.$ + +当变量增加到两个或更多时，多项式的唯一分解性不再成立。此时的情况更为有趣。理解它便是我们接下来的课题。 + +研究问题：将多元热带多项式分解为不可约热带多项式的结果并不唯一。这里有一个简单的例子： + +$(0 \odot x \oplus 0) \odot (0 \odot y \oplus 0) \odot (0 \odot x \odot y \oplus 0)$ + +$\qquad = (0 \odot x \odot y \oplus 0 \odot x \oplus 0) \odot (0 \odot x \odot y \oplus 0 \odot y \oplus 0).$ + +请开发一种算法（附实现与复杂度分析），用于计算给定热带多项式的所有不可约分解。高和劳德[8]已证明，热带分解对于经典意义上的多元多项式因式分解问题具有重要意义。 + +### 曲线 + +一个热带多项式函数 $p: \mathbb{R}^{n} \to \mathbb{R}$ 定义为有限个线性函数的最小值。我们将超曲面 $\mathcal{H}(p)$ 定义为所有满足以下条件的点 $\mathbf{x} \in \mathbb{R}^{n}$ 的集合：在该点处，该最小值至少由两个线性函数同时取得。等价地说，点 $\mathbf{x} \in \mathbb{R}^{n}$ 属于 $\mathcal{H}(p)$ 当且仅当 $p$ 在 $\mathbf{x}$ 处不是线性的。例如，若 $n = 1$ 且 \$\$为满足假设(2)的(1)式中的三次多项式，则 + +$\mathcal{H}(p) = \{b - a, c - b, d - c\}.$ + +因此，超曲面 $\mathcal{H}(p)$ 就是多项式 \$(x)\$的“根”的集合。 + +在本节中，我们考虑两个变量的多项式情形： + +$p(x, y) = \bigoplus_{(i,j)} c_{ij} \odot x^{i} \odot y^{j}.$ + +事实 3. 对于二元多项式 $p$，其对应的热带曲线 $\mathcal{H}(p)$是嵌入平面 $athbb{R}^{2}$的一个有限图。该图既包含有界边，也包含无界边，所有边的斜率均为有理数，并且图在每一个节点处均满足如下所述的零张力条件： + +考虑图的任意节点 $(x,y)$，不妨将其取为原点 $(0,0)$那么与该节点相邻的边均位于具有有理斜率的直线上。在从原点出发的每一条这样的射线上，取最小的非零格点向量。在 $(x,y)$的零张力即意味着这些向量的和为零。 + +我们的第一个例子是平面上的一条直线。它由以下多项式定义： + +$p(x,y) = a \odot x \oplus b \odot y \oplus c \qquad \mathrm{其中} \ a, b, c \in \mathbb{R}.$ + +曲线 $\mathcal{H}(p)$ 由所有使得函数 + +$p: \mathbb{R}^{2} \to \mathbb{R}, \qquad (x,y) \mapsto \min \bigl( a + x, \, b + y, \, c \bigr)$ + +非线性的点 $(x,y)$ 构成。它由从点 \$x,y) = (c - a, c - b)\$出发、分别指向正北、正东和西南方向的三个半射线组成。零张力条件体现为 $(0,0) + (0,1) + (-1,-1) = (0,0)$ + +下面是在平面上绘制热带曲线 $\mathcal{H}(p)$ 的一般方法。考虑出现在多项式 \$\$中的任意一项 \($amma \odot x^{i} \odot y^{j}\)$们用 $\thbb{R}^{3}$ 点 \((\$mma, i, j)\) 来$该项，并计算这些点在 \(\ma$bb{R}^{3}\) 中的$。然后，将该凸包的下包络通过映射 \(\mat$b{R}^{3} \to \mathbb{R}^{2}\)，即 \$\gamma, i, j) \mapsto (i, j)\)，投影到平$。所得图像是一个平面凸多边形，并带有一个划分 \(\Delt$)，将其细分为$的多边形。热带曲线 \(\mathc${H}(p)\)（实际上是它的$形式”）就是这个细分 \(\Delta\$的对偶图。回忆$，平面图的对偶图是另一个平面图，其顶点对应于原图的各个区域，而边则表示区域之间的相邻关系。 + +举一个例子，我们考虑一般二次多项式 + +$p(x,y) = a \odot x^{2} \oplus b \odot xy \oplus c \odot y^{2} \oplus d \odot x \oplus e \odot y \oplus f.$ + +此时，$\Delta$ 是以 \$0,0)\$\($,2)\)$ \(($0)\) $点的三角形的一个细分。格点 \((0$)\)、\$1,0)\)、\($,1$) 可被用$细分的顶点。假设 \(a, b$c, d, e, f \in \mathbb{R}\) 满足条件$ + +$2b \leq a + c, \quad 2d \leq a + f, \quad 2e \leq c + f,$ + +那么细分 $\Delta$ 由四个三角形、三条内部边和六条边界边构成。曲线 \$mathcal{H}(p)\$则有四个顶点、三条有界边和六条半射线（两条向北、两条向东、两条向西南）。在图 2 中，我们用带箭头的粗线标示了二次曲线 \($athcal{H}(p)\)$负形式。它是细分 \(\$lta\) $偶图，而 \(\D$ta\) 本$以细线绘出。 + +![](images/f1cf81ff5afe4d6be1ce08d3d84861c815510b2644387e6cd70efe6be95166cf.png) + +事实 4. 热带曲线的相交与插值性质与代数曲线类似。 + +1.两条一般直线交于一点，一条直线与一条二次曲线交于两点，两条二次曲线交于四点，依此类推。2. 过两个一般点有唯一一条直线，过五个一般点有唯一一条二次曲线，依此类推。 + +关于热带代数几何中贝祖定理的一般性讨论（见本刊封面图示），可参阅文献 [17]。 + +研究问题：对三维空间中度数为 $d$ 的热带曲线的所有组合类型进行分类。此类曲线是一个有限嵌入图，其形式为 + +$C = \mathcal{H}(p_{1}) \cap \mathcal{H}(p_{2}) \cap \dots \cap \mathcal{H}(p_{r}) \subset \mathbb{R}^{3},$ + +其中 $p_{i}$ 为热带多项式，\$\$在四个坐标方向上各有 \($)$无界平行半射线，而 \(C$ $有其余边均为有界。 + +## 系统发育学 + +计算生物学中的一个重要问题，是根据涉及 $n$ 个叶片的距离数据构建系统发育树。在生物学家的术语中，叶片的标签称为分类单元。这些分类单元可能是生物体或基因，每个均以 + +DNA 序列表示。关于系统发育学的入门，我们推荐 Felsenstein [7] 以及 Semple 和 Steele [18] 的著作。此处以 $n = 4$ 为例，说明此类数据如何产生。考虑四个基因组的序列比对： + +$ \text{人类:} ACAAATGTCATTAGCGAT\dots $ + +$ \text{小鼠:} ACGTTGTCAAATAGAGAT\dots $\$\text{大鼠:} ACGTAAGTCATTACACAT\dots \$[$text{鸡:} GCACAGTCAGTAGAGCT\dots \]$ + +从这类序列数据中，计算生物学家可推断任意两个分类单元之间的距离。有多种算法可执行此推断，它们皆基于进化的统计模型。就我们的讨论而言，可将任意两个字符串之间的距离视为汉明距离（即它们相异字符的比例）的精细化版本。在我们的（人类、小鼠、大鼠、鸡）示例中，推断出的距离矩阵可能为如下对称 $4\times 4$ 矩阵： + +$H M R C$ + +$H 0 1.1 1.0 1.4$\$ 1.1 0 0.3 1.3\$[$1.0 0.3 0 1.2\]$C$.4 1.3 1.2 0\]$ + +系统发育学的问题在于构建一棵树，使其边长能代表该距离矩阵，前提是这样的树存在。在我们的示例中，确实存在这样一棵树，如图3所示，其中每条边旁的数字为其长度。两个叶片之间的距离，是连接这两个叶片的唯一路径上各边长之和。例如，树中“人类”与“小鼠”之间的距离为 $0.6 + 0.3 + 0.2 = 1.1$，这正是 \$\times 4\$矩阵中的对应项。 + +![](images/82b70efc64eb5aae18a9a0b3a9243c5a70a345b3f49f3b997ee242fe6d34e59c.png) + +一般而言，考虑 $n$ 个分类单元，分类单元 \$\$与分类单元 \($)$间的距离为一个正实数 \(d$ij}\)，$种生物统计方法确定。因此，我们所得到的是一个实对称 \(n\$mes n\) 矩$ + +$D = \left( \begin{array}{ccccc}0 & d_{12} & d_{13} & \dots & d_{1n}\\ d_{12} & 0 & d_{23} & \dots & d_{2n}\\ d_{13} & d_{23} & 0 & \dots & d_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ d_{1n} & d_{2n} & d_{3n} & \dots & 0 \\ \end{array} \right).$ + +我们可假定 $D$ 为一度量，即对所有 \$, j, k\$三角形不等式 \(${ik} \leq d_{ij} + d_{jk}\)$成立。此性质可通过矩阵乘法表述如下： + +事实 5. 矩阵 $D$ 表示一个度量，当且仅当 \$ \odot D = D\$ + +若存在一棵具有 $n$ 片叶子的树 \$\$其叶子标记为 \($ 2, \ldots, n\)$ \(T$ $条边均被赋予正长度，使得任意两片叶子 \(i\$与$(j\) 之间$离恰为 \(d_{$}\)，则我们$义在 \(\{1,$, \ldots, n\}\) 上的度量$(D\) 为树度量。$量在生物学中天然出现，因为它们模拟了演化过程中产生 \(n\) 个$群的历$迹。 + +大多数度量 $D$ 并非树度量。若我们从生物数据中获得一个度量 \$\$则可合理假设存在一个与 \($)$近的树度量 \(D$\)。$学家运用多种算法（例如“邻接法”）从给定数据 \(D\$构$这样一棵邻近的树 \(T\)$文$述树度量的热带刻画。 + +令 $X = (X_{ij})$ 为一个对称矩阵，其对角线元素为零，而 \$binom{n}{2}\$个非对角元素为未知量。对于每个四元组 \($i, j, k, l\} \subset \{1, 2, \ldots, n\}\)$们考虑以下二阶热带多项式： + +$p_{ijkl} = X_{ij} \odot X_{kl} \oplus X_{ik} \odot X_{jl} \oplus X_{il} \odot X_{jk}. \quad (4)$ + +该多项式称为热带 Grassmann–Plücker 关系，它实质上是 $2 \times 4$ 矩阵中 \$ \times 2\$子行列式所满足的经典 Grassmann–Plücker 关系的热带版本 [14, 定理 3.20]。 + +它在空间 $\mathbb{R}^{\binom{n}{2}}$ 中定义了一个超曲面 \$mathcal{H}(p_{ijkl})\$热带 Grassmann 簇即是这 \($inom{n}{4}\)$超曲面的交集，记作 + +$G_{r2,n} = \bigcap_{1 \leq i < j < k < l \leq n} \mathcal{H}(p_{ijkl}).$ + +这一 $\mathbb{R}^{\binom{n}{2}}$ 的子集具有多面体扇的结构，即它由有限多个凸多面体锥拼接而成，且这些锥体之间衔接自然。 + +**事实 6.** 定义在 $\{1, 2, \ldots, n\}$ 上的度量 \$\$是树度量，当且仅当其负值 \($= -D\)$于热带 Grassmann 簇 \(G$r2,n}\) $ + +这一陈述等价于系统发育学中的四点条件：$D$ 为树度量当且仅当对所有 \$ \leq i < j < k < l \leq n\$三个数 \(${ij} + D_{kl}\)$(D_{ik} + D_{jl}\) $\(D$il} + D_{jk}\) 中$大值至少出现两次。令 \(X $-D\)，则意$三个数 \(X_{$} + X_{kl}\)、\(X$ik$+ X_{jl}\) 与 \(${il}$ X_{jk}\) 中的最小值$出现两次，亦即 \(X \in $athcal{H}(p_{ijkl})\)。热带 Gra$mann 簇 \(G_{r2,n$) 亦被称为系统发$的空间 [3, 14, 20]。这一优美空间的组合结构已得到充分研究与深刻理解。 + +通常，与其测量不同分类群之间的成对距离，从统计角度看，更精确的做法是考虑所有 $r$ 元组分类群，并联合度量每个 \$\$元组内的相异性。例如，在上述树中，三元组 {人类, 小鼠, 大鼠} 的联合相异性为 1.2，即包含小鼠、人类和大鼠的子树中所有边长之和。Lior Pachter 与第一作者在 [15] 中证明，只要 \($\geq 2r - 1\)$可从所有 \(r$ $的数据中重建出这棵树。 + +在2004年的原始讲义中，我们曾提出一个研究问题：刻画 $G_{r2,n}$ 到 \$mathbb{R}^{\binom{n}{r}}\$的给定嵌入之像，尤其针对 \($= 3\)$情形。此后，Christiano Bocci 与 Filip Cools [4] 已解决了 \(r$ 3\) $题，并在更高 \(r\$的$上取得了重要进展。尽管 + +仍有工作待完成，我们现在建议探讨以下一个较少被研究的问题，它取自 [14, 第3章] 的结尾。 + +研究问题 我们称一个度量 $D$ 具有系统发育秩 \$leq k\$若存在 \($)$树度量 \(D$(1)}, D^{(2)}, \ldots, D^{(k)}\)，$ + +$D_{i j} = \max \bigl( D_{i j}^{(1)}, D_{i j}^{(2)}, \ldots, D_{i j}^{(k)} \bigr) \qquad \text{对所有 } 1 \leq i, j \leq n.$ + +等价地，矩阵 $X = - D$ 是矩阵 \$^{(i)} = - D^{(i)}\$的和： + +$X = X^{(1)} \oplus X^{(2)} \oplus \dots \oplus X^{(k)}.$ + +系统发育秩这一概念旨在为混合了 $k$ 种不同进化历史的距离数据建模。系统发育秩 \$leq k\$的度量集合构成 \($athbb{R}^{\binom{n}{2}}\)$的一个多面体扇。请计算此扇，并探究其组合、几何与拓扑性质，特别是当 \(k$ 2\) $ + +## 热带线性空间 + +为推广直线的概念，我们定义热带超平面为 $\mathbb{R}^{n}$ 中形如 \$mathcal{H}(\ell)\$的子集，其中 \($ll\)$ \(n$ $知数的热带线性函数： + +$\ell(x) = a_{1} \odot x_{1} \oplus a_{2} \odot x_{2} \oplus \dots \oplus a_{n} \odot x_{n}.$ + +此处 $a_{1}, \ldots, a_{n}$ 为任意实常数。在热带数学中，解线性方程意味着计算有限多个超平面 \$mathcal{H}(\ell)\$的交集。人们很容易想将热带线性空间简单地定义为热带超平面的交集。然而，这并非一个好的定义，因为此类任意交集可能具有混合维数，且其行为方式不同于经典几何中的线性空间。 + +更好的热带线性空间概念源于只允许那些“足够完备”的超平面交集，其含义我们稍后会说明。我们给出的定义直接推广了此前关于系统发育学的讨论。其核心思想是：系统发育树即热带射影空间中的直线，其 Plücker 坐标 $X_{i j}$ 为两两距离 \$_{i j}\$的负值。 + +考虑 $\binom{n}{d}$ 维空间 \$mathbb{R}^{\binom{n}{d}}\$其坐标 \(${i_{1} \dots i_{d}}\)$ \(\$,2,\ldots ,n\}\) $\(d\$元$ \(\{i$1},\ldots ,i_{d}\}\) 为索$令 \(S\) $\${1,2,\ldots ,n\}\) 的任意 $(d-2)\) 元子集，并$\(i, j,$, l\) 为 \(\{$\ld$s ,n\} \setminus S\) 中任意四个互异$。相应的三项 Grassmann-Plücker 关系 \(p_{S,i $k l}\) 是如下二次热带多$： + +$p_{S,i j k l} = X_{S i j} \odot X_{S k l} \oplus X_{S i k} \odot X_{S j l} \oplus X_{S i l} \odot X_{S j k}. \quad (5)$ + +我们定义 Dressian 为如下交集 + +$D_{r d,n} = \bigcap_{S,i,j,k,l} \mathcal{H}(p_{S,i j k l}) \subset \mathbb{R}^{\binom{n}{d}},$ + +其中交集遍历所有满足上述条件的 $S$、\$\$\($)$(k\)、$l\)。术$Dressian”源于代数学家安德烈亚斯·德雷斯（Andreas Dress），他目前从事计算生物学研究。有关其工作的相关参考文献及更多细节，请参阅文献[11]。 + +注意到在特殊情况 $d = 2$ 下，我们有 \$ = \emptyset\$此时多项式(5)即为(4)中的四点条件。在此特殊情形中，\($r_{2,n} = G r_{2,n}\)$正是前文讨论过的系统发育树空间。 + +现在，我们在 Dressian $D_{r_{d,n}}$ 中取定一个坐标为 \$X_{i_{1} \dots i_{d}})\$的任意点 \($)$于 \(\$, 2, \dots, n\}\) $意 \((d$)\)-子$\(\{j$0}, j_{1}, \dots, j_{d}\}\)，考虑$关于变量 \(x_{1$ \dots, x_{n}\) 的热带$形式： + +$\ell_{j_{0} j_{1} \dots j_{d}}^{X} = \bigoplus_{r=0}^{d} X_{j_{0} \dots \widehat{j_{r}} \dots j_{d}} \odot x_{r}, \quad (6)$ + +其中符号 $\widehat{\cdot}$ 表示省略 \$_{r}\$与点 \($)$关联的热带线性空间定义为如下集合： + +$L_{X} = \bigcap \mathcal{H}(\ell_{j_{0} j_{1} \dots j_{d}}^{X}) \subset \mathbb{R}^{n}.$ + +这里交集遍历 $\{1, 2, \dots, n\}$ 的所有 \$d+1)\$子集 \($j_{0}, j_{1}, \dots, j_{d}\}\)$ + +热带线性空间正是形如 $L_{X}$ 的集合，其中 \$\$为 \(${r_{d,n}} \subset \mathbb{R}^{\binom{n}{d}}\)$的任意一点。这些对象在文献[21]和[11]中有详细研究。本节第一段所提及的“充分完备性”，意指我们需要利用上述 \(L$X}\) $式来求解线性方程，以确保超平面的交集确实构成一个线性空间。此处给出的线性空间定义比文献[6, 17, 20]中使用的定义更具包容性——在那些文献中，要求 \(L_$}\) 必$源于带有适当赋值的域上的经典代数几何。 + +例如，$\mathbb{R}^{n}$ 中的一个三维热带线性子空间（亦即热带射影 \$n-1)\$空间中的二维平面），是 \($inom{n}{4}\)$热带超平面的交集，其中每个定义线性形式均含四项： + +$\ell_{j_{0} j_{1} j_{2} j_{3}}^{X} = X_{j_{0} j_{1} j_{2}} \odot x_{j_{3}} \oplus X_{j_{0} j_{1} j_{3}} \odot x_{j_{2}} \oplus X_{j_{0} j_{2} j_{3}} \odot x_{j_{1}} \oplus X_{j_{1} j_{2} j_{3}} \odot x_{j_{0}}.$ + +值得注意的是，即便在 $X$ 的每个坐标均为 \$\$乘法单位元）或 \($nfty\)$法单位元）这一非常特殊的情形下，所得结构仍十分有趣。此时，\(L$X}\) $个多面体扇，称为拟阵的伯格曼扇[1]。 + +热带线性空间具备许多普通线性空间的性质。首先，它们是具有正确维数的纯多面体复形： + +事实 7. 热带线性空间 $L_{X}$ 的每个极大胞腔都是 \$\$维的。 + +每个热带线性空间 $L_{X}$ 都唯一地决定了其热带Plücker坐标向量 \$\$这种唯一性仅差一个公共标量的热带乘法（即经典加法）。若 \($)$ \(L$\prime}\) $为维数 \(d\$和$(d^{\prime}\) 的热$性空间，且满足 \(d +$^{\prime}\geq n\)，则 \$\) 与 \(${\p$me}\) 必然相交。$来说，两个热带线性空间的交集并不一定是热带线性空间，但几乎总是如此。具体而言，若 \(L\) 和$(L^$pri$}\) 是维数分别为 $d\) 和 \(d^{\$im$\) 的热带线性空间，且$(d + d^{\prime}\geq n\)，而 \(v\) 是一$般$小向量，$ \(L\cap (L^$prime} + v)\) 就是一个维数为 \(d $d^{\pri$} - n\) 的热带线性空间。依照文献[$]，我们可以合理地定义 \(L\) 和 \(L^{$rime}$ $定交：取 \(L\cap $^{\pri$} + v)\) 在 \(v\) 趋于零时的极限，$限$是一个维$ \(d + d^{\prime} $n\) 的热带线性空间。$ + +一个 $d$ 维热带线性空间并不总能表示为 \$ - d\$个热带超平面的交。从定义可知，\($inom{n}{d+1}\)$超平面总是足够的。在2004年的原始讲义中，我们曾提出这样的问题：在 \(n$ $间中，要刻画任意一个 \(d\$维$线性空间，最少需要多少个热带超平面？\(n\)$超$是否总是足够？这些问题已由Tristram Bogart在[2, Theorem 2.10]中给出解答，而Josephine Yu和Debbie Yuster在[22]中提供了更精细的组合分析。本文不再提出新的研究问题，而是以一个问题作结。 + +是否存在关于热带几何的教科书？截至2009年6月，尽管其基本定义具有初等性，但似乎仍缺乏热带几何的入门教材。 + +目前唯一已出版的热带代数几何专著是卷[10]，该书基于2004年由Ilia Itenberg、Grigory Mikhalkin和Eugenii Shustin在Oberwolfach举办的研讨会。该书着重探讨了热带几何与拓扑及实代数几何的联系。此外，多篇综述文章从不同角度提供了入门指引。除[17]外，我们特别推荐Andreas Gathmann[9]和Eric Katz[12]的阐述，这些内容主要面向具备代数几何背景的读者。Grigory Mikhalkin目前正在为Clay数学研究所丛书撰写一部热带几何研究专著，而Diane Maclagan与第二作者也已启动名为《热带几何导论》的书籍项目，其初稿可从作者主页获取。2009年秋季，伯克利数学科学研究所（MSRI）将举办一个关于热带几何的特别学期。 + +致谢。Speyer在此工作中得到Clay数学研究所研究奖学金的资助，Sturmfels则获得美国国家科学基金会（DMS-0456960、DMS-0757236）的资助。 + +### 参考文献 + +1. 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