Tropical Algebraic Geometry

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line MCS 563 Lecture 39 Analytic Symbolic Computation Jan Verschelde, 21 April 2014 Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 1 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 2 / 34

logarithms of algebraic sets Denoting C = C \ {0}, we consider the application of to a variety. log : C C R R (x, y) (log( x ), log( y )) Introduced by Gelfand, Kapranov, and Zelevinsky, log(v) for a variety V is called the amoeba of a variety. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 3 / 34

plot for a line For a line, we use polar coordinates to plot the amoeba: f := 1 2 x + 1 5 y 1 = 0, A := [ ln ( re Iθ ), ln ( 5 2 reiθ 5 )]. > f := 1/2*x + 1/5*y - 1: > s := solve(f,y): > L := map(log,map(abs,[x,s])): > A := subs(x=r*exp(i*theta),l); > Ap := seq(plot([op(subs( theta=k*pi/200,a)),r=-100..100], thickness=6),k=0..99): > plots[display](ap,axes=none); The amoeba of a linear polynomial with Maple. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 4 / 34

compactifying the amoeba We compactify the amoeba of f 1 (0): Take lines perpendicular to the tentacles. As each line cuts the plane in half, keep those halves of the plane where the amoeba lives. The intersection of all half planes defines a polygon. The resulting polygon is the Newton polygon of f. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 5 / 34

amoeba and Newton polygon The edges of the Newton polygon are perpendicular to the tentacles of the amoeba. 12 10 8 6 4 2 (0,1) (0,0) (1,0) K2 0 2 4 6 8 10 12 K2 Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 6 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 7 / 34

normal cones and fan The inner product is, : Z 2 Z 2 Z : ((i, j),(u, v)) iu + jv. Let P be the Newton polygon of the polynomial f. The normal cone to a vertex p of P is { v R 2 \ {0} p, v = min q, v }. q P The normal cone to an edge spanned by p 1 and p 2 is { v R 2 \ {0} p 1, v = p 2, v = min q, v }. q P The normal fan of P is the collection of all normal cones to vertices and edges of P. Given f, a tropicalization of f, denoted by Trop(f), is a finite collection of inner normals (u, v) (normalized as gcd(u, v) = 1) to the edges of the Newton polygon P of f. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 8 / 34

a quadratic polynomial Consider a regular triangulation of the Newton polygon of a general quadratic polynomial. Take the normals to the facets of the upper hull: Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 9 / 34

intersecting two quadrics Consider two general quadratic polynomials: We see a tropical version of the theorem of Bézout. This tropical version is an interpretation of the computation of the mixed cells in a regular mixed-cell configuration via the common refinement of the normal fans to their edges. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 10 / 34

a dictionary An analogy between classical algebraic geometry and tropical algebraic geometry: classical tropical C,+, polynomial zero set hypersurface toric variety irreducible algebraic variety R, min,+ piecewise linear function singular locus normal fan ordinary linear variety balanced (n 1)-skeleton Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 11 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 12 / 34

a railway network Consider two stations: S 1 and S 2 ; three circuits: one inner between the two stations and two outer circuits, serving the suburbs. Four trains travel back and forth. S 1 S 2 5 2 3 3 Numbers along the tracks are fixed travel times. A good timetable obey the rules: 1 the frequency of trains should be as high as possible; 2 the frequency of trains is the same along all tracks; 3 trains wait to allow the changeover of passengers; 4 the trains depart at a station as soon as allowed. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 13 / 34

inequalities Denote by x 1 and x 2 the common departure times of the trains respectively at S 1 and S 2. The initial departure times are x 1 (0) and x 2 (0) and the kth departure times are given by x 1 (k 1) and x 2 (k 1). The rules above translate into the inequalities: x 1 (k + 1) = max(x 1 (k) + 2, x 2 (k) + 5) x 2 (k + 1) = max(x 1 (k) + 3, x 2 (k) + 3) In the max-plus algebra, replacing max by and addition by, we write x 1 (k + 1) = (x 1 (k) 2) (x 2 (k) 5) x 2 (k + 1) = (x 1 (k) 3) (x 2 (k) 3) Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 14 / 34

tropical linear algebra In matrix-vector form: x(k + 1) = A x(k), x(k) = ( x1 (k) x 2 (k) ) ( 2 5, A = 3 3 Eigenvalues λ and eigenvectors v of a matrix A are defined in the max-plus algebra as A v = λ v, where not all components of v are equal to. ). Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 15 / 34

tropical eigenvalues Eigenvectors are (similar to conventional linear algebra) not uniquely determined. If the same constant is added to all components of an eigenvector, then we get again an eigenvector. It is typical to normalize an eigenvector so its first component is zero. For the solution of the system above, we write x(1) = A x(0) = λ x(0) and in general x(k) = λ k x(0), k = 1, 2,... Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 16 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 17 / 34

curves of cyclic 4-roots x 0 + x 1 + x 2 + x 3 = 0 x f(x) = 0 x 1 + x 1 x 2 + x 2 x 3 + x 3 x 0 = 0 x 0 x 1 x 2 + x 1 x 2 x 3 + x 2 x 3 x 0 + x 3 x 0 x 1 = 0 x 0 x 1 x 2 x 3 1 = 0 One tropism v = (+1, 1,+1, 1) with in v (f)(z) = 0: x 1 + x 3 = 0 x in v (f)(x) = 0 x 1 + x 1 x 2 + x 2 x 3 + x 3 x 0 = 0 x 1 x 2 x 3 + x 3 x 0 x 1 = 0 x 0 x 1 x 2 x 3 1 = 0. We look for solutions of the form (x 0 = t +1, x 1 = z 1 t 1, x 2 = z 2 t +1, x 3 = z 3 t 1 ). Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 18 / 34

solving the initial form system Substitute (x 0 = t +1, x 1 = z 1 t 1, x 2 = z 2 t +1, x 3 = z 3 t 1 ): in v (f)(x 0 = t +1, x 1 = z 1 t 1, x 2 = z 2 t +1, x 3 = z 3 t 1 ) (1 + z 2 )t +1 = 0 z = 1 + z 1 z 2 + z 2 z 3 + z 3 = 0 (z 1 z 2 + z 3 z 1 )t +1 = 0 z 1 z 2 z 3 1 = 0. We find two solutions: (z 1 = 1, z 2 = 1, z 3 = +1) and (z 1 = +1, z 2 = 1, z 3 = 1). Two space curves ( t, t 1, t, t 1) and ( t, t 1, t, t 1) satisfy the entire cyclic 4-roots system. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 19 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 20 / 34

Asymptotics of Witness Sets Consider a witness sets for an algebraic space curve. A witness set for a degree d curve contains d points on a general hyperplane c 0 + c 1 x 1 + c 2 x 2 + + c n x n = 0, satisfying f(x) = 0. We then deform a witness set for a curve in two stages: 1 The first homotopy moves to a hyperplane in special position: { h (1) (x, t) = f(x) = 0, for t from 1 to 0 : (c 0 + c 1 x 1 + + c n x n )t + (c 0 + c 1 x 1 )(1 t) = 0. 2 After renaming c 0 + c 1 x 1 = 0 into x 1 = γ, we let x 1 go to zero with the following homotopy: { h (2) f(x) = 0 (x, t) = x 1 γt = 0, for t from 1 to 0. The two homotopies need further study. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 21 / 34

the first homotopy In moving a general plane into special position, some paths will diverge, consider for example f(x 1, x 2 ) = x 1 x 2 1. Lemma All solutions at the end of the homotopy h (1) (x, t) = 0 lie on the curve defined by f(x) = 0 and in the hyperplane x 1 = c 0 /c 1. Proof. We claim that we find the same solutions to h (1) (x, t = 0) = 0 either by using the homotopy { h (1) f(x) = 0, for t from 1 to 0 : (x, t) = (c 0 + c 1 x 1 + + c n x n )t + (c 0 + c 1 x 1 )(1 t) = 0. or by solving h (1) (x, 0) = 0 directly. This claim follows from cheater s homotopy or the more general coefficient-parameter polynomial continuation. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 22 / 34

the second homotopy For simplest example of the hyperbola x 1 x 2 1 = 0: its solution is (x 1 = t, x 2 = t 1 ) and the tropism is v = (1, 1). Lemma As t 0 in the homotopy h (2) (x, t) = 0, the leading powers of the Puiseux series expansions are the components of a tropism. In particular, the expansions have the form { x1 = t x k = c k t v k(1 + O(t)), k = 2,..., n. (1) Proof. Following Bernshteǐn s second theorem, a solution at infinity is a solution in (C ) n of an initial form system. For a solution to have values in (C ) n, all equations in that system need to have at least two monomials. So the system is an initial form system defined by a tropism. To arrive at the form of (1) for the solution of h (2) (x, t) = 0 we rescale the parameter t so we may replace x 1 = γt by x 1 = t. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 23 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 24 / 34

tropical monomials Let x 1, x 2,..., x n be variables representing elements in (R { },, ), then a tropical monomial is a product of variables, allowing repetition. Example: x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 = x 2 1 x 3 2 x 2 3 x 4. As a function, a tropical monomial is a linear function. x 2 + x 1 + x 3 + x 1 + x 4 + x 2 + x 3 + x 2 = 2x 1 + 3x 2 + 2x 3 + x 4. Every linear function in n variables with integer coefficients we can write as a tropical monomial, exponents can be negative. Tropical monomials are the linear functions on R n with integer coefficients. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 25 / 34

tropical polynomials Any finite linear combination of tropical monomials defines a tropical polynomial with real coefficients and integer exponents: p(x 1, x 2,...,x n ) = c a1 x a 1,1 1 x a 1,2 2 x a 1,n n The corresponding function: c a2 x a 2,1 1 x a 2,2 2 x a 2,n n c ak x a k,1 1 x a k,2 2 x a k,n n. p(x 1, x 2,..., x n ) = min( c a1 + a 1,1 x 1 + a 1,2 x 2 + + a 1,n x n, c a2 + a 2,1 x 1 + a 2,2 x 2 + + a 2,n x n,..., c ak + a k,1 x 1 + a k,2 x 2 + + a k,n x n ). Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 26 / 34

piecewise-linear functions Tropical polynomials as functions p : R n R 1 are continuous; 2 are piecewise-linear, with a finite number of pieces; and 3 are concave, i.e.: p ( x+y ) 2 1 2 (p(x) + p(y)) for all x, y R. Any function which satisfies these three properties can be represented as the minimum of a finite set of linear functions. Lemma The tropical polynomials in n variables are the piecewise-linear functions on R n with integer coefficients. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 27 / 34

the graph of a tropical quadratic polynomial a x 2 b x c = min(a + 2x, b + x, c) a + 2x b + x c b b a roots c If b a c b, then p(x) = a (x (b a)) (x (c b)) = a + min(x, b a) + min(x, c b). Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 28 / 34

the tropical fundamental theorem of algebra Theorem Every tropical polynomial function can be written uniquely as a tropical product of tropical linear functions. Note: Distinct polynomials can represent the same function. x 2 17 x 2 = min(2x, x + 17, 2) = min(2x, x + 1, 2) = x 2 1 x 2 = (x 1) 2 2x 1 + x 2 Every polynomial can be replaced by an equivalent polynomial (equivalent means: representing the same function) that can be factored into linear functions. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 29 / 34

Tropical Algebraic Geometry 1 Amoebas an asymptotic view on varieties tropicalization of a polynomial 2 The Max-Plus Algebra at Work making time tables for a railway network 3 Polyhedral Methods for Algebraic Sets solving the cyclic 4-roots problem asymptotics of witness sets 4 Tropical Polynomials the tropical fundamental theorem of algebra graphing a tropical line Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 30 / 34

graph of the tropical line L(x, y) = 1 x 2 y 3 y = 1 z x x = 2 y = 2 (2,1,3) (2,1) y x = 1 ( 1, 2) y = x 1 L : R 2 R (x, y) min(1 + x, 2 + y, 3) z = 1 + x z = 2 + y z = 1 + x z = 3 z = 2 + y z = 3 } : y = x 1 } : x = 2 } : y = 1 Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 31 / 34

a picture in the plane L : R 2 R : (x, y) min(1 + x, 2 + y, 3) Look where the minimum is attained at least twice: y x = 2 y = x 1 y = 1 x Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 32 / 34

intersecting two tropical lines Intersecting 1 x 2 y 3 with 2 x 4 y 2: where min(1+x, 2+y, 3) and min( 2+x, 4+y, 2) attain their mininum twice. y x = 4 x = 2 y = x 1 y = x 6 y = 1 x y = 2 Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 33 / 34

Summary + Exercises Tropical algebraic geometry provides a framework to apply polyhedral methods to solve polynomial systems. Exercises: 1 Consider the amoeba of the product of two linear equations. How many tentacles to you see in each direction? For which choices of the coefficients do you see holes in the amoeba? 2 Prove the tropical fundamental theorem of algebra. 3 Investigate the complexity of graphing a tropical plane curve of degree d and express the complexity as a function of d. Analytic Symbolic Computation (MCS 563) Tropical Algebraic Geometry L-39 21 April 2014 34 / 34