The Fundamental Theorem of Tropical Geometry Elizabeth Kelley Abstract This paper focuses on understanding the statement of the fundamental theorem of tropical geometry through concrete examples. In order to do so, we again use examples to explicitly build up the necessary background knowledge, including: tropicalization, valuations, tropical hypersurfaces, splittings, initial forms, and tropical varities. We finish by briefly commenting on the overall structure of the proof of the fundamental theorem. Tropical geometry is a relatively recent field. We might think of it as a variant on algebraic geometry, where we choose to work over the tropical semi-ring, but it also has strong connections to other fields like com- binatorics, enumerative and real algebraic geometry, mathematical physics, number theory, symplectic ge- ometry, and (surprisingly!) computational biology, among others. In this paper, we will be most concerned with its connections to commutative algebra, as our main focus will be on understanding the statement of the fundamental theorem of tropical algebraic geometry - which will require an understanding of topics like tropical varieties. In that vein, we need to start by building the requisite background for the fundamental theorem. Through- out, we will work in the tropical semi-ring (R [ f¥g, ⊕, ), where ⊕ and are defined as x ⊕ y := min(x, y) and x y := x + y. These operations have all the nice properties that we might expect - both are associative and commutative, have identity elements (i.e., x ⊕ ¥ = x and x 0 = x for all x 2 R), and obey a distributive law. As one might expect, it’s possible to construct and study polynomials in the tropical semi-ring. Let x1,..., xn be elements of (R [ f¥g, ⊕, ).A tropical monomial in these elements is any product of x1,..., xn, where repetition is allowed, an example being: 3 2 x1 x2 x1 x4 x3 x2 x1 = x1x2x3x4 (Note that because is commutative, we can collect like terms and use our usual product shorthand). If we evaluated this example using classical arithmetic, it would become x1 + x1 + x1 + x2 + x2 + x3 + x4 = 3x1 + 2x2 + x3 + x4. which we recognize as a linear function from R4 to R. More generally, any tropical monomial is a lin- ear function from Rn to R. We can construct tropical polynomials as finite linear combinations of tropical monomials. For example, 2 ax1x2 ⊕ bx3x4 ⊕ cx1x4 where a, b, c 2 R is a tropical polynomial mapping R4 to R. Note that this map is not necessarily linear, though; if we translate to classical arithmetic, this tropical polynomial is equivalent to writing minfa + 2x1 + x2, b + x3 + x4, c + x1 + x4g, More generally, any tropical polynomial is classically equivalent to taking the minimum of a finite set of linear functions. An example is shown in Figure 1. y = 2x y = 1 y = x Figure 1: The graph of the tropical polynomial x2 ⊕ x ⊕ 1. Another useful notion is that of tropicalization, where we pass classical functions to the tropical semi-ring. Tropicalization allows us to reframe problems in the language of tropical geometry. Since the notion of tropicalization is not entirely straightforward, we need to begin by establishing some background material. First, we need to introduce valuations. Let K be a field, with K∗ denoting the non-zero elements of K. Then a valuation on K is defined as a functional val : K ! R [ f¥g such that: val(a) = ¥ if and only if a = 0; ∗ val(ab) = val(a) + val(b); and val(a + b) ≥ minfval(a), val(b)g for all a, b 2 K . We typically use Gval to denote the image of val. To illustrate the idea of a valuation, consider the field of Puiseux series, kfftgg - a generalization of power series which allows exponents of indeterminates to be negative and rational - with coefficients in an alge- a a a braically closed field of characteristic zero. Elements of kfftgg have the form c(t) = c1t 1 + c2t 2 + c3t 3 + ··· ∗ where ci 2 k and a1 < a2 < a3 < ··· are a strictly increasing set of rational numbers with a common de- nominator. We can then define the natural valuation on kfftgg as val : kfftgg ! R, where val maps c(t) 2 kfftgg to the lowest exponent in its series representation (in our expression for c(t), this would be a1). To get a feel for how this valuation works, one might verify the following pair of examples: 6t4 + 7t2 7 val = val 2t3 + t = 1, 3t 3 val (7) = 0. Here we have Gval = Q, since we can construct an element c(t) 2 kfftgg with lowest exponent correspond- ing to any given rational number. Although we will work more generally with a generic field K, the field of Puiseux series appears with some frequency in the study of tropical geometry. Now that we’ve established the concept of a valuation, we’re ready to explore the tropicalization of a poly- u ±1 ±1 ±1 nomial. Let f (x) = ∑ cux be a Laurent polynomial in K[x1 , x2 ,..., xn ]. Then its tropicalization, trop( f ) : Rn+1 ! R is given by n+1 trop( f )(w) = minfval(cu) + w · u : u 2 N , cu 6= 0g, which is obtained from f (x) by sending constant coefficients to their valuation and then replacing classical addition and multiplication with the tropical operations. As an example, let K be the field of Puiseux series with real coefficients and consider −3/2 3 8 2 4 11 −3 f (x) = (5t + 7t + 9t )x1 + (7t + 10t )x2 ±1 ±1 in K[x1 , x2 ]. Using the previously defined natural valuation on the Puiseux series, we observe that −3/2 3 8 4 11 −1 −1 −1 trop( f )(w) = minfval(5t + 7t + 9t ) ⊕ w1 ⊕ w1, val(7t + 10 ) ⊕ w2 ⊕ w2 ⊕ w2 ⊕g −1 −1 −1 = minf−3/2 ⊕ w1 ⊕ w1, 4 ⊕ w2 ⊕ w2 ⊕ w2 g −1 = minf−3/2 + 2w1, 4 + 3w2 g, 2 for w = (w1, w2) 2 R . ±1 ±1 ±1 We already have a notion of the classical algebraic variety of a Laurent polynomial f 2 K[x1 , x2 ,..., xn ] as a hypersurface of (K∗)n, written V( f ) = fx 2 (K∗)n : f (x) = 0g. The same polynomial f 2 I also has an associated tropical hypersurface, denoted as trop(V( f )), which consists of the set of points where the minimum of trop( f ) is not unique. In order to rigorously define this, we need to first introduce the idea of initial forms and splittings. ∗ u u A splitting f : Gval ! K is a group homomorphism sending u 7! t such that val(t ) = u. In our running example, K = kfftgg and val : c(t) 7! lowest exponent of c(t), a natural choice of splitting is u 7! tu. If we instead let K = C and use the trivial valuation (val(a) = 0 for all a 6= 0), a natural choice might be 0 7! 1. Let R = fa 2 K : val(a) ≥ 0g be the valuation ring of K, which is local and has maximal ideal m = fa 2 K : val > 0g [ f0g. Then let k = R/m be the resulting residue field. For K = kfftgg, we have 1/n k R = [n≥1k[[t ]] and = k. For a in the valuation ring R, we denote its image in k as a. Similarly, for a polynomial f with coefficients in R, we use f to denote the polynomial formed by replacing each coefficient with its image in the residue field. n+1 The initial form of f is defined relative to some fixed w 2 (Gval) . If we let W = trop( f )(w) = minfval(cu) + w · u : cu 6= 0g, then we can define the initial form of f relative to w as −W w w inw( f ) = t f (t 0 x0,..., t n xn) −W w·u u = t ∑ cut x u2Nn+1 w·u−W u = ∑ cut x u2Nn+1 −val(c ) u = ∑ cut u x u2Nn+1 val(cu)+w·u=W −val(c ) u = ∑ cut u x u2Nn+1 val(cu)+w·u=W 2 2 4 ±1 ±1 ±1 As an example, consider f = (t + t )x0 + 2t x1 + 3t x2 2 Cfftgg[x0 , x1 , x2 ] and let w = (2, 1, 0). Then W = minfval(t + t2) + 2, val(2t2) + 1, val(3t4) + 0g = minf3, 3, 4g = 3, −3 2 −3 2 2 2 4 in(2,1,0)( f ) = t f (t x0, tx1, x2) = t ((t + t )t x0 + (2t )tx1 + 3t x2) = 1 + tx0 + 2x1 + 3tx2 = x0 + 2x1. (0, val(a+1)) min = w1 = 0 (val(a), 0) min = w2 = 0 (val(a), val(a)) min = w1, w2 (a) The graph of (b) The graph of the closure of (c) The graph of the closure of 2 trop( f ) = minfx, y, 0g val(X) fw 2 Gval : inw(I) 6= h1ig More formally, the tropical hypersurface associated with f is the closure (in the Euclidean topology) in Rn ±1 ±1 of fw 2 Gval : hinw( f )i 6= h1ig, i.e. the set of weight vectors for which inw( f ) isn’t a unit in k[x1 ,..., xn ]. ±1 ±1 ±1 Tropicalization of functions allows us to elegantly define tropical varieties. Let I be an ideal in K[x1 , x2 ,..., xn ] with algebraic variety X = V(I). Then the tropicalization of X, trop(X), is given by the intersection of the tropical hypersurfaces defined by each f 2 I. I.e., \ trop(X) = trop(V( f )) f 2I An important and perhaps not immediately obvious thing to note is that it’s not sufficient to just intersect the hypersurfaces corresponding to some generating set for I.