Mixed Discriminants

MIXED DISCRIMINANTS EDUARDO CA TT ANI, MAR ´ IA ANG ´ ELICA CUETO, ALICIA DICKENSTEIN, SANDRA DI R OCCO, AND BERND STURMFELS Abstract. The mixed discriminan t of n Lauren t p olynomials in n v ariables is the irre- ducible polynomial in the co efficien ts which v anishes whenever t w o of the ro ots coincide. The Ca yley tric k expresses the mixed discriminant as an A -discriminan t. W e sho w that the degree of the mixed discriminan t is a piecewise linear function in the Pl ¨ uc k er co ordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves. De dic ate d to the memory of our friend Mikael Passar e (1959–2011) 1. Introduction A fundamen tal topic in mathematics and its applications is the study of systems of n p olynomial equations in n unkno wns x = ( x 1 , x 2 , . . . , x n ) o v er an algebraically closed field K : (1.1) f 1 ( x ) = f 2 ( x ) = · · · = f n ( x ) = 0 . Here we consider Lauren t p olynomials with fixed supp ort sets A 1 , A 2 , . . . , A n ⊂ Z n : (1.2) f i ( x ) = X a ∈ A i c i,a x a ( i = 1 , 2 , . . . , n ) . If the co efficien ts c i,a are generic then, according to Bernstein ’s The or em [ 3 ], the n umber of solutions of ( 1.1 ) in the algebraic torus ( K ∗ ) n equals the mixe d volume MV ( Q 1 , Q 2 , . . . , Q n ) of the Newton p olytop es Q i = con v( A i ) in R n . Ho wev er, for sp ecial c hoices of the co ef- ficien ts c i,a , t w o or more of these solutions come together in ( K ∗ ) n and create a p oint of higher m ultiplicit y . The conditions under whic h this happ ens are enco ded in an irreducible p olynomial in the co efficients, whose zero lo cus is the variety of il l-p ose d systems [ 18 , § I-4]. While finding this polynomial is usually b eyond the reac h of sym b olic computation, it is often p ossible to describ e some of its inv arian ts. Our aim here is to c haracterize its degree. An isolated solution u ∈ ( K ∗ ) n of ( 1.1 ) is a non-de gener ate multiple r o ot if the n gradient v ectors ∇ x f i ( u ) are linearly dep endent, but an y n − 1 of them are linearly independent. This condition means that u is a regular p oint on the curv e defined b y any n − 1 of the equations in ( 1.1 ). W e define the discriminantal variety as the closure of the locus of co efficien ts c i,a for whic h the asso ciated system ( 1.1 ) has a non-degenerate multiple ro ot. If the discriminan tal v ariety is a h yp ersurface, we define the mixe d discriminant of the system ( 1.1 ) to b e the unique (up to sign) irreducible polynomial ∆ A 1 ,...,A n with in teger coefficients in the unkno wns c i,a whic h defines it. Otherwise w e say that the system is defe ctive and set ∆ A 1 ,...,A n = 1. In the non-defectiv e case, w e ma y iden tify ∆ A 1 ,...,A n with an A -discriminant in the sense of Gel’fand, Kapranov and Zelevinsky [ 12 ]. A is the Cayley matrix ( 2.1 ) of A 1 , . . . , A n . This 2010 Mathematics Subje ct Classific ation. 13P15, 14M25, 14T05, 52B20. Key wor ds and phr ases. A-discriminan t, degree, multiple ro ot, Ca yley p olytop e, tropical discriminant, matroid strata, mixed Grassmannian. 1 2 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS matrix has as columns the v ectors in the lifted configurations e i × A i ∈ Z 2 n for i = 1 , . . . , n . The relationship b etw een ∆ A 1 ,...,A n and the A -discriminant will b e made precise in Section 2 . In Section 3 we fo cus on the case n = 2. Here, the mixed discriminant ∆ A 1 ,A 2 expresses the condition for t wo plane curv es { f 1 = 0 } and { f 2 = 0 } to b e tangen t at a common smo oth p oin t. In Theorem 3.3 we present a general form ula for the bidegree ( δ 1 , δ 2 ) of ∆ A 1 ,A 2 . In nice sp ecial cases, to b e describ ed in Corollary 3.15 , that formula simplifies to (1.3) δ 1 = area( Q 1 + Q 2 ) − area( Q 1 ) − p erim( Q 2 ) , δ 2 = area( Q 1 + Q 2 ) − area( Q 2 ) − p erim( Q 1 ) , where Q i is the con vex hull of A i and Q 1 + Q 2 is their Mink owski sum. The ar e a is normalized so that a primitiv e triangle has area 1. The p erimeter of Q i is the cardinality of ∂ Q i ∩ Z 2 . The formula ( 1.3 ) applies in the classical case, where f 1 and f 2 are dense p olynomials of degree d 1 and d 2 . Here, ∆ A 1 ,A 2 is the classical tact invariant [ 17 , § 96] whose bidegree equals (1.4) ( δ 1 , δ 2 ) =  d 2 2 + 2 d 1 d 2 − 3 d 2 , d 2 1 + 2 d 1 d 2 − 3 d 1  . See Benoist [ 2 ] and Nie [ 15 ] for the analogous form ula for n dense p olynomials in n v ari- ables. The righ t-hand side of ( 1.3 ) is alw a ys an upp er b ound for the bidegree of the mixed discriminan t, but in general the inequalit y is strict. Indeed, consider t wo sparse p olynomials (1.5) f 1 ( x 1 , x 2 ) = c 10 + c 11 x d 1 1 + c 12 x d 1 2 and f 2 ( x 1 , x 2 ) = c 20 + c 21 x d 2 1 + c 22 x d 2 2 , with d 1 and d 2 p ositiv e coprime integers, then the bidegree drops from ( 1.4 ) to (1.6)  d 2 2 + 2 d 1 d 2 − 3 d 2 · min { d 1 , d 2 } , d 2 1 + 2 d 1 d 2 − 3 d 1 · min { d 1 , d 2 }  . In Section 4 w e pro ve that the degree of the mixed discriminan t, in the natural Z n -grading, is a piecewise p olynomial function in the co ordinates of the p oints in A 1 , A 2 , . . . , A n . Theorem 1.1. The de gr e e of the mixe d discriminant cycle is a pie c ewise line ar function in the Pl¨ ucker c o or dinates on the mixe d Gr assmannian G (2 n, I ) . It is line ar on the tr opic al matr oid str ata of G (2 n, I ) determine d by the c onfigur ations A 1 , . . . , A n . The formula on e ach maximal str atum is unique mo dulo the line ar forms on ∧ 2 n R m that vanish on G (2 n, I ) . Here, the cycle refers to the mixed discriminan t raised to a p o w er that expresses the index in Z n of the sublattice affinely spanned b y A 1 ∪ · · · ∪ A n . The mixed Grassmannian G (2 n, I ) parameterizes all 2 n -dimensional subspaces of R m that arise as row spaces of Cayley matri- ces ( 2.1 ) with m = P n i =1 | A i | columns, and I is the partition of { 1 , . . . , m } sp ecified by the n configurations. This Grassmannian is regarded as a sub v ariet y in the exterior p o w er ∧ 2 n R m , via the Pl ¨ uc ker em b edding by the maximal minors of the matrix ( 2.1 ). See Definition 4.2 for details. The mixed Grassmannian admits a combinatorial stratification in to tr opic al matr oid str ata , and our assertion says that the degree of the mixed discriminan t cycle is a p olyno- mial on these strata. The pro of of Theorem 1.1 is based on tr opic al algebr aic ge ometry , and sp ecifically on the combinatorial construction of the tropical discriminant in [ 7 ]. MIXED DISCRIMINANTS 3 2. Ca yley Configura tions Let A 1 , . . . , A n b e configurations in Z n , defining Lauren t p olynomials as in ( 1.2 ). W e shall relate the mixed discriminant ∆ A 1 ,...,A n to the A -discriminant, where A is the Ca yley matrix (2.1) A = Ca y ( A 1 , . . . , A n ) =       1 0 · · · 0 0 1 · · · 0 . . . . . . . . . . . . 0 0 · · · 1 A 1 A 2 · · · A n       . This matrix has 2 n ro ws and m = P n i =1 | A i | columns, so 0 = (0 , . . . , 0) and 1 = (1 , . . . , 1) denote row vectors of appropriate lengths. W e introduce n new v ariables y 1 , y 2 , . . . , y n and enco de our system ( 1.1 ) b y one auxiliary p olynomial with supp ort in A , via the Cayley trick : φ ( x, y ) = y 1 f 1 ( x ) + y 2 f 2 ( x ) + · · · + y n f n ( x ) . W e denote by ∆ A the A -discriminant as defined in [ 12 ]. That is, ∆ A is the unique (up to sign) irreducible p olynomial with integer co efficien ts in the unkno wns c i,a whic h v anishes whenev er the hypersurface { ( x, y ) ∈ ( K ∗ ) 2 n : φ ( x, y ) = 0 } is not smo oth. Equiv alen tly , ∆ A is the defining equation of the dual v ariet y ( X A ) ∨ when this v ariety is a hypersurface. Here, X A denotes the pro jectiv e toric v ariety in P m − 1 asso ciated with the Cayley matrix A . If ( X A ) ∨ is not a h yp ersurface, then no such unique p olynomial exists. W e then set ∆ A = 1 and refer to A as a defe ctive configuration. It is useful to keep trac k of the lattice index i ( A ) = i ( A, Z 2 n ) =  Z 2 n : Z · A  , where Z · A is the Z -linear span of the columns of A . The discriminant cycle is the polynomial ˜ ∆ A = ∆ i ( A ) A . The same construction mak es sense for the mixed discriminan t and it results in the mixe d discriminant cycle ˜ ∆ A 1 ,...,A n . The exp onents i ( A ) will be compatible in the following theorem. Theorem 2.1. The mixe d discriminant e quals the A -discriminant of the Cayley matrix: ∆ A 1 ,...,A n = ∆ A . This result is more subtle than it may seem at first glance. It implies that ( A 1 , . . . , A n ) is defectiv e if and only if A is defectiv e. The t w o discriminantal v arieties can differ in that case. Example 2.2. Let n = 2 and consider the Cayley matrix A =   1 0 0 1 A 1 A 2   =     1 1 1 0 0 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 2     The corresp onding system ( 1.1 ) consists of tw o univ ariate quadrics in different v ariables: f 1 ( x 1 ) = c 10 + c 11 x 1 + c 12 x 2 1 = 0 and f 2 ( x 2 ) = c 20 + c 21 x 2 + c 22 x 2 2 = 0 . This system cannot hav e a non-degenerate m ultiple ro ot, for any choice of coefficients c ij , so the ( A 1 , A 2 )-discriminan tal v ariet y is empt y . On the other hand, the A -discriminantal v ariet y is non-empty . It has co dimension t w o and is defined by c 2 11 − 4 c 10 c 12 = c 2 21 − 4 c 20 c 22 = 0. ♦ 4 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS Pr o of of The or em 2.1 . W e ma y assume i ( A ) = 1. Let u ∈ ( K ∗ ) n b e a non-degenerate m ultiple root of f 1 ( x ) = · · · = f n ( x ) = 0. Our definition ensures the existence of a unique (up to scaling) vector v ∈ ( K ∗ ) n suc h that P n i =1 v i ∇ x f i ( u ) is the zero vector. The pair ( u, v ) ∈ ( K ∗ ) 2 n is a singular p oin t of the hypersurface defined by φ ( x, y ) = 0. By pro jecting into the space of co efficients c i,a , we see that the ( A 1 , . . . , A n )-discriminan tal v ariet y is con tained in the A -discriminantal v ariety . Example 2.2 shows that this containmen t can b e strict. W e no w claim that ∆ A 6 = 1 implies ∆ A 1 ,...,A n 6 = 1. This will establish the proposition b ecause ∆ A 1 ,...,A n is a factor of ∆ A , and ∆ A is irreducible, so the tw o discriminants are equal. Eac h p oint ( u, v ) ∈ ( K ∗ ) 2 n defines a p oin t on X A . If ∆ A 6 = 1, the dual v ariet y ( X A ) ∨ is a h yp ersurface in the dual pro jective space ( P m − 1 ) ∨ . Moreo v er, see e.g. [ 13 ], a generic h yp erplane in the dual v ariety is tangent to the toric v ariet y X A at a single p oint. Consider a generic p oint on the conormal v ariety of X A in P m − 1 × ( P m − 1 ) ∨ . It is represen ted b y a pair  ( u, v ) , c  , where ( u, v ) ∈ ( K ∗ ) 2 n and c is the co efficient vector of a p olynomial φ ( x, y ) such that ( u, v ) is the unique singular p oin t on { φ ( x, y ) = 0 } . The co efficient vector c defines a p oin t on the ( A 1 , . . . , A n )-discriminan tal v ariety unless we can relab el such that the gradients of f 1 , . . . , f n − 1 are linearly dep endent at u . Assuming that this holds, we let n − 1 X i =1 t i ∇ x f i ( u ) = 0 b e the dep endency relation and set t = ( t 1 , . . . , t n − 1 , 0) 6 = 0 . The p oin t  ( t + u, v ) , c  lies on the conormal v ariety of X A . This implies that the generic hyperplane defined by c is tangen t to X A at t wo distinct points ( u, v ) 6 = ( t + u, v ), whic h cannot happ en. It follo ws that ∆ A 1 ,...,A n 6 = 1, as we w an ted to show. This concludes our pro of.  Example 2.3. Let n = 2 and A 1 = A 2 = { (0 , 0) , (1 , 0) , (0 , 1) , (1 , 1) } , a unit square. Then f 1 ( x 1 , x 2 ) = a 00 + a 10 x 1 + a 01 x 2 + a 11 x 1 x 2 , f 2 ( x 1 , x 2 ) = b 00 + b 10 x 1 + b 01 x 2 + b 11 x 1 x 2 . The Cayley configuration A is the standard 3-dimensional cub e. The A -discriminant is kno wn to b e the hyp er determinant of format 2 × 2 × 2, by [ 12 , Chapter 14], whic h equals ∆ A 1 ,A 2 = a 2 00 b 2 11 − 2 a 00 a 01 b 10 b 11 − 2 a 00 a 10 b 01 b 11 − 2 a 00 a 11 b 00 b 11 +4 a 00 a 11 b 01 b 10 + a 2 01 b 2 10 + 4 a 01 a 10 b 00 b 11 − 2 a 01 a 10 b 01 b 10 − 2 a 01 a 11 b 00 b 10 + a 2 10 b 2 01 − 2 a 10 a 11 b 00 b 01 + a 2 11 b 2 00 . Theorem 2.1 tells us that this h yp erdeterminan t coincides with the mixed discriminant of f 1 and f 2 . Note that the bidegree equals ( δ 1 , δ 2 ) = (2 , 2), and therefore ( 1.3 ) holds. ♦ W e no w shift gears and fo cus on defectiv e configurations. W e kno w from Theorem 2.1 that ( A 1 , . . . , A n ) is defectiv e if and only if the asso ciated Cayley configuration A is defectiv e. While there has b een some recen t progress on c haracterizing defectiveness [ 7 , 10 , 14 ], the problem of classifying defective configurations A remains op en, except in cases when the c o dimension of A is at most four [ 6 , 8 ] or when the toric v ariety X A is smo oth or Q -factorial [ 4 , 9 ]. Recall that X A is smo oth if and only if, at each ev ery v ertex of the p olytop e Q = con v ( A ), the first elemen ts of A that lie on the inciden t edge directions form a basis for the lattice spanned by A . The v ariety X A is Q -factorial when Q is a simple p olytop e, that is, when ev ery v ertex of Q lies in exactly dim( Q ) facets. Note that smo oth implies Q -factorial. MIXED DISCRIMINANTS 5 W e set dim( A ) = dim( Q ), and w e sa y that A is dense if A = Q ∩ Z d . A subset F ⊂ A is called a fac e of A , denoted F ≺ A , if F is the intersection of A with a face of the p olytop e Q . W e will denote by s n the standard n -simplex and b y σ n the configuration of its vertic es . When A is the Ca yley configuration of A 1 , . . . , A n ⊂ Z n , the co dimension of A is m − 2 n . This n um b er is usually rather large. F or instance, if all n p olytop es Q i = conv( A i ) are full-dimensional in R n then co dim( A ) > n · ( n − 1), and th us, for n > 3, w e are outside the range where defectiv e configurations ha ve b een classified. How ev er, if n = 2 and the configurations A 1 and A 2 are full-dimensional we can classify all defectiv e configurations. Prop osition 2.4. L et A 1 , A 2 ⊂ Z 2 b e ful l-dimensional c onfigur ations. Then, ( A 1 , A 2 ) is defe ctive if and only if, up to affine isomorphism, A 1 and A 2 ar e b oth tr anslates of p · σ 2 , for some p ositive inte ger p . Pr o of. Let A = Cay( A 1 , A 2 ). Both A 1 and A 2 app ear as faces of A . In order to pro v e that A is non-defectiv e, it suffices to exhibit a 3-dimensional non-defectiv e sub configuration (see [ 5 , Prop osition 3.1] or [ 10 , Proposition 3.13]). Let u 1 , u 2 , u 3 b e non-collinear p oints in A 1 and v 1 , v 2 distinct p oints in A 2 . The subconfiguration { u 1 , u 2 , u 3 , v 1 , v 2 } of A is 3-dimensional and non-defective if and only if no four of the p oin ts lie in a h yp erplane or, equiv alen tly , if the vector v 2 − v 1 is not parallel to an y of the v ectors u j − u i , j 6 = i . W e can alw ays find suc h sub configurations unless A 1 and A 2 are the vertices of triangles with parallel edges. In the latter case, w e can apply an affine isomorphism to get A 1 = p · σ 2 and A 2 a translate of ± q · σ 2 , where p and q are p ositiv e in tegers. The total degree of the mixed discriminan t equals deg(∆ p · σ 2 ,q · σ 2 ) = ( p 2 + q 2 + pq − 3 min { p, q } 2 ) / gcd( p, q ) 2 , deg(∆ p · σ 2 , − q · σ 2 ) = ( p + q ) 2 / gcd( p, q ) 2 . The first formula follo ws from ( 1.6 ) and it is p ositive unless p = q . The second form ula will b e deriv ed in Example 3.10 . It alw ays gives a positive num b er. This concludes our proof.  Similar arguments can be used to study the case when one of the configurations is one- dimensional. Ho wev er, it is more instructiv e to classify suc h defectiv e configurations from the bidegree of the mixed discriminant. This will b e done in Section 3 . Corollary 2.5. L et A 1 and A 2 b e ful l-dimensional c onfigur ations in Z 2 . Then the mixe d discriminantal variety of ( A 1 , A 2 ) is either a hyp ersurfac e or empty. R emark 2.6 . The same result holds in n dimensions when the toric v ariet y X A is smo oth and A 1 , . . . , A n are full-dimensional configurations in Z n . Under these hypotheses, ( A 1 , . . . , A n ) is defective if and only if each A i is affinely equiv alen t to p · σ n , with p ∈ N . In particular, the mixed discriminan tal v ariet y of ( A 1 , . . . , A n ) is either a h yp ersurface or empty . The “if ” direction is straightforw ard: w e may assume i ( A ) = 1 and p = 1 by replacing Z n with the lattice spanned by pe 1 , . . . , pe n . Then, the system ( 1.1 ) consists of linear equations, and it is clearly defectiv e. The “only-if ” direction is deriv ed from results in [ 9 ]: ( A 1 , . . . , A n ) is defectiv e if and only if the (2 n − 1)-dimensional polytop e Q = con v ( A ) is isomorphic to a Ca yley polytop e of at least t + 1 ≥ n + 1 configurations of dimension k < t that hav e the same normal fan. As we already hav e a Cayley structure of n configurations in dimension n , w e deduce t = n and k = n − 1. Then, w e should hav e Q ' s n − 1 × s n ' s n × s n − 1 . After an affine transformation, all n p olytop es Q i are standard n -simplices and all A i are translates of s n . This shows that A has an ”in verted” Cayley structure of n + 1 copies of σ n − 1 . 6 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS W e expect Proposition 2.4 to hold in n dimensions without the smoothness h yp othesis in Remark 2.6 . Clearly , whenever the mixed volume of Q 1 , . . . , Q n is 1, then there are no m ultiple roots and we hav e ∆ A 1 ,...,A n = 1. The follo wing result giv es a necessary and sufficien t condition for b eing in this situation: up to affine equiv alence, this is just the linear case. Prop osition 2.7. If A 1 , . . . , A n ar e n -dimensional c onfigur ations in Z n then the mixe d vol- ume MV ( Q 1 , . . . , Q n ) is 1 if and only if, up to affine isomorphism, A 1 = · · · = A n = σ n . Pr o of. W e shall prov e the “only-if ” direction by induction on n . Supp ose MV( Q 1 , Q 2 , . . . , Q n ) = 1. By the Aleksandrov-F enc hel inequality , we hav e v ol( Q i ) = 1 for all i , where the volume form is normalized so that the standard n -simplex has v olume 1. Since the mixed v olume function is monotone in each co ordinate, for an y choice of edges l i in Q i w e hav e 0 6 MV ( l 1 , l 2 , . . . , l n ) 6 MV ( l 1 , Q 2 , . . . , Q n ) 6 MV ( Q 1 , . . . , Q n ) = 1 . Since all p olytop es Q i are full-dimensional, w e can pick n linearly indep enden t edges l 1 , . . . , l n . Therefore MV( l 1 , . . . , l n ) > 0 and MV ( l 1 , l 2 , . . . , l n ) = MV ( l 1 , Q 2 , . . . , Q n ) = 1. In particular, the edge l 1 has length one. After a change of co ordinates w e may assume that l 1 = e n , the n -th standard basis v ector. Denote b y π the pro jection of Z n on to Z n / Z · e n ' Z n − 1 and the corresp onding map of R -vector spaces. W e then hav e MV ( π ( Q 2 ) , . . . , π ( Q n )) = 1. By the induction hypothesis, w e can transform the first n − 1 co ordinates so that π ( A 2 ) = · · · = π ( A n ) = σ n − 1 . This means that A i ⊂ σ n − 1 × Z · e n . Now, let a i b e a p oint in A i not lying in the co ordinate hyperplane x n = 0. Then 1 6 vol(con v ( σ n − 1 , a i )) 6 v ol( Q i ) = 1, and w e conclude that Q i = con v ( σ n − 1 , a i ). But, since vol( Q i ) = 1, it follows that a i = b i ± e n , for some b i ∈ σ n − 1 . By repeating this pro cess with an edge of A 1 con taining the p oint a 1 , w e see that all b i ’s are equal and that the sign of e n in all a i ’s is the same. This shows that, after an affine isomorphism, we hav e A 1 = · · · = A n = σ n , yielding the result.  3. Two Cur ves in the Plane In this section w e study the condition for t w o plane curv es to b e tangen t. This condition is the mixed discriminan t in the case n = 2. Our primary goal is to pro ve Theorem 3.3 , whic h giv es a form ula for the bidegree of the mixed discriminan t cycle of t wo full-dimensional planar configurations A 1 and A 2 . Remark 3.11 addresses the degenerate case when one of the A i is one-dimensional. Our main to ol is the connection betw een discriminan ts and principal determinan ts. In order to make this connection precise, and to define all the terms appearing in ( 3.3 ), we recall some basic notation and facts. W e refer to [ 10 , 12 ] for further details. Let A ⊂ Z d and Q the con vex hull of A . As is customary in toric geometry , w e assume that A lies in a rational h yp erplane that do es not pass through the origin. This holds for Ca yley configurations ( 2.1 ). Giv en any subset B ⊂ A w e denote b y Z · B , resp ectively R · B , the linear span of B ov er Z , resp ectively ov er R . F or any face F ≺ A we define the lattic e index i ( F , A ) :=  R · F ∩ Z d : Z · F  . W e set i ( A ) = i ( A, A ) = [ Z d : Z · A ]. W e consider the A -discriminant ∆ A and the princip al A -determinant E A . They are defined in [ 12 ] under the assumption that i ( A ) = 1. If i ( A ) > 1 then we change the ambien t lattice from Z d to Z · A , and w e define the asso ciated cycles ˜ E A = E i ( A ) A and ˜ ∆ A = ∆ i ( A ) A . The expressions on the right-hand sides are computed relative to the lattice Z · A . MIXED DISCRIMINANTS 7 R emark 3.1 . The principal A -determinan t of [ 12 , Chapter 10] is a polynomial E A in the v ariables c α , α ∈ A . Its Newton p olytop e is the se c ondary p olytop e of A , and its degree is ( d + 1) v ol(conv( A )), where v ol = vol Z · A is the normalized lattice v olume for Z · A . W e alwa ys ha ve deg( ˜ E A ) = ( d + 1) vol Z d (con v ( A )), where v ol Z d is the normalized lattice volume for Z d . W e state the factorization form ula of Gel’fand, Kaprano v and Zelevinsky [ 12 , Theorem 1.2, Chapter 10] for the principal A -determinan t as in Estero v [ 10 , Prop osition 3.10]: (3.1) ˜ E A = ± ˜ ∆ A · Y F ≺ A ˜ ∆ u ( F,A ) F . The pro duct runs o ver all prop er faces of A . The exp onents u ( F , A ) are computed as follo ws. Let π denote the pro jection to R · A/ R · F and Ω the normalized volume form on R · A/ R · F . This form is normalized with resp ect to the lattice π ( Z d ), so that the fundamental domain with resp ect to integer translations has volume (dim( R · A ) − dim( R · F ))!. W e set u ( F , A ) := Ω  con v ( π ( A )) \ conv( π ( A \ F ))  . R emark 3.2 . The p ositive integers u ( F , A ) are denoted c F,A in [ 10 ]. If i ( A ) = 1 then u ( F , A ) is the sub diagr am volume asso ciated with F , as in [ 12 , Theorem 3.8, Chapter 5]. W e now sp ecialize to the case of Cayley configurations A = Ca y ( A 1 , A 2 ), where A 1 , A 2 ⊂ Z 2 are full-dimensional. Here, A is a 3-dimensional configuration in the h yp erplane x 1 + x 2 = 1 in R 4 . Note that i ( A, Z 4 ) = i ( A 1 ∪ A 2 , Z 2 ). The configurations A 1 and A 2 are facets of A . W e sa y that F is a vertic al face of A if F ≺ A but F 6≺ A i , i = 1 , 2. The v ertical facets of A are either triangles or tw o-dimensional Ca yley configurations defined by edges e ≺ A 1 and f ≺ A 2 . This happ ens if e and f are parallel and hav e the same orientation, that is, if they hav e the same in ward normal direction when view ed as edges in Q 1 and Q 2 . W e call suc h edges str ongly p ar al lel and denote the vertical facet they define by V ( e, f ). Let E i denote the set of edges of A i and set P = { ( e, f ) ∈ E 1 × E 2 : e is strongly parallel to f } . W e write ` ( e ) for the normalize d length of an edge e with respect to the lattice Z 2 . F or v ∈ A 1 w e define (3.2) mm( v ) = MV ( Q 1 , Q 2 ) − MV(conv( A 1 \ v ) , Q 2 ) , and similarly for v ∈ A 2 . This quantit y is the mixe d multiplicity of v in ( A 1 , A 2 ). Theorem 3.3. L et A 1 and A 2 b e ful l-dimensional c onfigur ations in Z 2 . Then δ 1 := deg A 1 ( ˜ ∆ A 1 ,A 2 ) = area( Q 2 ) + 2 MV( Q 1 , Q 2 ) − X ( e,f ) ∈P min { u ( e, A 1 ) , u ( f , A 2 ) } ` ( f ) − X v ∈ V ert A 1 mm( v ) . (3.3) δ 2 := deg A 2 ( ˜ ∆ A 1 ,A 2 ) = area( Q 1 ) + 2 MV( Q 1 , Q 2 ) − X ( e,f ) ∈P min { u ( e, A 1 ) , u ( f , A 2 ) } ` ( e ) − X v ∈ V ert A 2 mm( v ) . 8 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS Theorem 3.3 is the main result in this section. W e shall deriv e it from the following form ula (whic h is immediate from ( 3.1 )) for the bidegree of our mixed discriminant: (3.4) bideg ( ˜ ∆ A 1 ,A 2 ) = bideg( ˜ E A ) − 2 X k =1 X F ≺ A k u ( F , A ) bideg( ˜ ∆ F ) − X F ≺ A vertical u ( F , A ) bideg( ˜ ∆ F ) . Note the need for the cycles ˜ ∆ A 1 ,A 2 , ˜ E A and ˜ ∆ F in this formula. W e shall pro ve Theorem 3.3 b y studying each term on the righ t-hand side of ( 3.4 ), one dimension at a time. A series of lemmas facilitates the exp osition. Lemma 3.4. The bide gr e e bideg( E A ) of the princip al A -determinantal cycle ˜ E A e quals (3.5) (3 area( Q 1 ) + area( Q 2 ) + 2 MV( Q 1 , Q 2 ) , area( Q 1 ) + 3 area( Q 2 ) + 2 MV( Q 1 , Q 2 )) . Pr o of. By [ 12 ], the total degree of ˜ E A is 4 vol( Q ). F rom any triangulation of A we can see v ol( Q ) = area( Q 1 ) + area( Q 2 ) + MV( Q 1 , Q 2 ) . Examining the tetrahedra in a triangulation rev eals that the bidegree is given b y ( 3.5 ).  An y pyramid is a defective configuration; hence, the v ertical facets of Q that are triangles do not con tribute to the right-hand side of ( 3.4 ) and can b e safely ignored from no w on. In particular, w e see that the only non-defective v ertical facets are the trap ezoids V ( e, f ) for ( e, f ) ∈ P . The following lemma explains their contribution to ( 3.4 ). Lemma 3.5. L et V ( e, f ) b e the vertic al fac et of A asso ciate d with ( e, f ) ∈ P . Then (1) bideg ( ˜ ∆ V ( e,f ) ) = ( ` ( f ) , ` ( e )) , (2) u ( V ( e, f ) , A ) = min { u ( e, A 1 ) , u ( f , A 2 ) } . Pr o of. The configuration V ( e, f ) is the Ca yley lift of tw o one-dimensional configurations. Its discriminan tal cycle is the resultant of t w o univ ariate polynomials of degree ` ( e ) and ` ( f ), so (1) holds. In order to prov e (2), we note that u ( V ( e, f ) , Q ) equals the normalized length of a segment in R 3 / R · V ( e, f ) starting at the origin and ending at the pro jection of a p oin t in A 1 or A 2 . This image is the closest p oin t to the origin in the line generated by the pro jection of Q . Thus, the m ultiplicity u ( V ( e, f ) , A ) is the minimum of u ( e, A 1 ) and u ( f , A 2 ).  W e next study the horizon tal facets of A given by A 1 and A 2 . Lemma 3.6. The discriminant cycle of the plane curve define d by A i has total de gr e e deg( ˜ ∆ A i ) = 3 area( Q i ) − X e ∈ Edges A i u ( e, A i ) deg( ˜ ∆ e ) − X v ∈ V ert A i u ( v , A i ) , wher e u ( v , A i ) = area( Q i ) − area(con v ( A i \ v )) . Pr o of. This is a special case of ( 3.4 ) b ecause deg ( ˜ E A i ) = 3 · area( Q i ) and deg ( ˜ ∆ v ) = 1 for an y vertex v ∈ A i . The statement ab out u ( v , A i ) is just its definition.  Next, w e consider the edges of A . The v ertical edges are defectiv e since they consist of just tw o p oints. Thus w e need only examine the edges of A 1 and A 2 . Lemma 3.7. L et e b e an e dge of A i . Then u ( e, A ) = u ( e, A i ) . MIXED DISCRIMINANTS 9 Pr o of. Consider the pro jection π : Q i → R 2 / R · e . The image π ( Q i ) is a segment of length M 1 = max { ` ([0 , π ( m )]) : m ∈ A i } , while conv( A i \ e ) pro jects to a segment of length M 2 = max { ` ([0 , π ( m )]) : m ∈ ( A i \ e ) } . Th us u ( e, A i ) = M 1 − M 2 . Next, consider the pro jection Q → R 3 / R · e . The images of A and con v( A \ e ) under this pro jection are trapezoids in R 3 / R · e . Their set-theoretic difference is a triangle of heigh t 1 and base M 2 − M 1 .  Lemma 3.8. L et v b e a vertex of A i . Then u ( v , A ) = u ( v , A i ) + mm( v ) . Pr o of. Supp ose v ∈ A 1 . The volume form Ω is normalized with resp ect to the lattice Z 3 . The volume of our Cayley p olytop e Cay( A 1 , A 2 ) equals area( Q 1 ) + area( Q 2 ) + MV( Q 1 , Q 2 ), and the analogous form ula holds for con v ( A \ v ) = Cay( A 1 \ v , A 2 ). W e conclude u ( v , A ) = vol(Ca y ( A 1 , A 2 )) − v ol(Cay( A 1 \ v , A 2 )) = area( Q 1 ) − area(con v ( A 1 \ v ) + MV( Q 1 , Q 2 ) − MV(conv( A 1 \ v ) , Q 2 )) = u ( v , A 1 ) + mm( v ) .  Pr o of of The or em 3.3 . By symmetry , it suffices to prov e ( 3.3 ). W e start with the A 1 -degree of the principal A -determinan tal cycle ˜ E A giv en in ( 3.5 ). In light of ( 3.1 ), we subtract the A 1 -degrees of the v arious discriminant cycles corresp onding to all faces of A . Besides the con tribution from A 1 , having u ( A 1 , A ) = 1 and given by Lemma 3.6 , only the vertices and the vertical facets contribute. Using Lemmas 3.7 and 3.8 , w e deriv e the desired form ula.  A t this p oint, the reader ma y find it an instructive exercise to deriv e ( 1.4 ) and ( 1.6 ) from Theorem 3.3 , and ditto for ( δ 1 , δ 2 ) = (2 , 2) in Example 2.3 . Here are t wo further examples. Example 3.9. Let A 1 and A 2 b e the dense triangles ( d 1 s 2 ) ∩ Z 2 and ( − d 2 s 2 ) ∩ Z 2 . Here, i ( A ) = 1 and ˜ ∆ A 1 ,A 2 = ∆ A 1 ,A 2 . W e hav e MV ( d 1 s 2 , − d 2 s 2 ) = 2 d 1 d 2 and P = ∅ . Computation of the mixed areas in ( 3.2 ) yields mm( v ) = d 2 for v ertices v ∈ A 1 and mm( v ) = d 1 for v ertices v ∈ A 2 . W e conclude bideg(∆ A 1 ,A 2 ) =  d 2 2 + 4 d 1 d 2 − 3 d 2 , d 2 1 + 4 d 1 d 2 − 3 d 1  . ♦ Example 3.10. Let A 1 = d 1 σ 2 and A 2 = − d 2 σ 2 . This is the sparse v ersion of Exam- ple 3.9 . Now, i ( A ) = g 2 , where g = gcd( d 1 , d 2 ), and ˜ ∆ A 1 ,A 2 = ∆ g 2 A 1 ,A 2 . W e still ha v e MV( d 1 s 2 , − d 2 s 2 ) = 2 d 1 d 2 and P = ∅ , but mm( v ) = d 1 d 2 for all v ∈ A . Hence bideg(∆ A 1 ,A 2 ) = 1 g 2  d 2 2 + d 1 d 2 , d 2 1 + d 1 d 2  . ♦ R emark 3.11 . F rom ( 3.4 ) we may also deriv e form ulas for the bidegree of the mixed dis- criminan t in the case when one of the configurations, sa y A 2 , is one-dimensional. The main differences with the pro of of Theorem 3.3 is that now we must treat A 2 as an edge, rather than a facet, and that it is enough for an edge e of Q 1 to b e parallel to Q 2 in order to hav e a non-defectiv e v ertical facet of Q . Clearly , there are at most tw o possible edges of Q 1 parallel to Q 2 . The A 2 -degree of the mixed discriminant cycle no w has a v ery simple expression: (3.6) δ 2 = area( Q 1 ) − X e || Q 2 u ( e, A 1 ) ` ( e ) , where the sum runs ov er all edges e of Q 1 whic h are parallel to Q 2 . In particular, if no edge of Q 1 is parallel to Q 2 , then δ 2 > 0 and ( A 1 , A 2 ) is not defectiv e. If only one edge e of Q 1 is parallel to Q 2 then δ 2 = 0 if and only if area( Q 1 ) = u ( e, A 1 ) ` ( e ) but this happ ens only if there is a single p oin t of A 1 not lying in the edge e . This means that 10 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS A = Ca y ( A 1 , A 2 ) is a p yramid and hence is defective. Finally , if there are tw o edges e 1 and e 2 of Q 1 parallel to Q 2 then δ 2 = 0 if and only if area( Q 1 ) = u ( e 1 , A 1 ) ` ( e 1 ) + u ( e 2 , A 1 ) ` ( e 2 ). This can only happen if all the p oints of A 1 lie either in e 1 or e 2 . In this case, A is the Ca yley lift of three one-dimensional configurations, and it is defective as w ell. Our next goal is to provide a sharp geometric b ound for the sum of the mixed multiplicities. W e start by pro viding a metho d to compute such in v ariants b y means of mixed subdivisions. Lemma 3.12. L et A 1 , A 2 b e ful l-dimensional in Z 2 and v ∈ A 1 . Any mixe d sub division of Q ∗ = con v( A 1 \ v ) + Q 2 extends to a mixe d sub division of Q = Q 1 + Q 2 . The mixe d multiplicity mm( v ) is the sum of the Euclide an ar e as of the mixe d c el ls in the closur e D of Q \ Q ∗ . v a 3 a 0 a 1 a 2 b 0 b 1 b 2 v + b 0 v + b 1 v + b 2 a 3 + b 2 a 0 + b 0 a 1 + b 0 a 2 + b 0 a 2 + b 1 a 2 + b 2 Figure 1. Geometric computation of the mixed multiplicit y mm( v ) via a suitable mixed sub division of the tw o polygons. The region D is sho wn in grey . Pr o of. Let E 0 2 ( v ) denote the collection of edges in A 2 whose inner normal directions lie in the r elative interior of the dual cone to a vertex v of A 1 . Equiv alen tly , E 0 2 ( v ) consists of those edges [ b, b 0 ] of A 2 suc h that v + b and v + b 0 are b oth vertices of A 1 + A 2 . See Figure 1 . First, assume E 0 2 ( v ) = ∅ . Then, there exists a unique b ∈ A 2 suc h that v + b is a vertex of Q . It follo ws that there exist a 0 , . . . , a r ∈ A 1 suc h that D is a union of triangles of the form (3.7) D = r [ i =1 con v ( { v + b, a i − 1 + b, a i + b } ) , and { a i − 1 + b, a i + b } are edges in the sub division of Q ∗ . Then, w e can extend the sub division of Q ∗ b y adding the triangles in ( 3.7 ). This do es not c hange the mixed areas and mm( v ) = 0. Supp ose no w that E 0 2 ( v ) = { f 1 , . . . , f s } , s > 1, with indices in coun terclo ckwise order. Let b 0 , . . . , b s b e the v ertices of A 2 suc h that f i is the segmen t [ b i − 1 , b i ]. The pairs v + b i − 1 , v + b i , for i = 1 , . . . , s , define edges of Q which lie in the b oundary of D . Let a 0 , a r ∈ A 1 b e the vertices of the edges of Q 1 adjacen t to v . W e insert r − 1 p oin ts in A 1 to form a coun terclo c kwise oriented sequence a 0 , a 1 , . . . , a r of vertices of conv( A 1 \ v ). Then a 0 + b 0 and a r + b s are vertices of Q ∗ , and the b oundary of D consists of the s segmen ts [ v + b i − 1 , v + b i ], together with segments of the form [ a i − 1 + b j , a i + b j ] or [ a i + b j − 1 , a i + b j ]. Figure 1 depicts MIXED DISCRIMINANTS 11 the case r = 3 , s = 2. Given this data, we sub divide D into mixed and unmixed cells. The unmixed cells are triangles { v + b j , a i +1 + b j , a i + b j } coming from the edges [ a i + b j , a i +1 + b j ] of Q ∗ . The mixed cells are parallelograms { v + b j , v + b j +1 , a i + b j +1 , a i + b j } built from the edges [ a i + b j , a i + b j +1 ] of Q ∗ . This sub division is compatible with that of Q ∗ .  W e write E 0 1 for the set of all edges of A 1 that are not strongly parallel to an edge of A 2 . The set E 0 2 is defined analogously . Prop osition 3.13. L et A 1 , A 2 ∈ Z 2 b e two-dimensional c onfigur ations. Then (1) The sum of the lengths of al l e dges in the set E 0 j is a lower b ound for the sum of the mixe d multiplicities over al l vertic es of the other c onfigur ation A i . In symb ols, X v ∈ V ert A i mm( v ) > X e ∈E 0 j ` ( e ) for j 6 = i. (2) If E 0 j = ∅ then X v ∈ V ert A i mm( v ) = 0 . (3) If i ( A 1 ) = i ( A 2 ) = 1 and the thr e e toric surfac es c orr esp onding to A 1 , A 2 and A 1 + A 2 ar e smo oth then the b ound in (1) is sharp. Pr o of. W e k eep the notation of the pro of of Lemma 3.12 . Recall that the set E 0 2 is the union of the sets E 0 2 ( v ), where v runs o v er all v ertices in A 1 . By Lemma 3.12 , the mixed m ultiplicity mm( v ) is the sum of the Euclidean areas of the mixed cells in D . Each mixed cell is a parallelogram { v + b k − 1 , v + b k , a i + b k − 1 , a i + b k } , so its area is ` ([ b k − 1 , b k ]) · ` ([ v , a i ]) · | det( τ k − 1 , η i ) | , where τ k − 1 and η i are primitive normal vectors to the edges [ b k , b k +1 ] and [ v , a i ]. Th us mm( v ) > P e ∈E 0 2 ( v ) ` ( e ). Since E 0 2 is the disjoint union of the sets E 0 2 ( v ), summing o v er all vertices v of A 1 giv es the desired lo w er bound. Part (2) also follows from Lemma 3.12 , as E 0 2 = ∅ implies that the sub division of D has no mixed cells. It remains to prov e (3). The assumption that X A 1 is smo oth implies that the segmen t [ a 0 , a 1 ] is an edge in con v( A 1 \ v ). Therefore, all mixed cells in D are parallelograms with v ertices { v + b k − 1 , v + b k , a i + b k − 1 , a i + b k } for i = 0 , 1, k = 1 , . . . , s . This parallelogram has Euclidean area | det( a i − v , b k − b k − 1 ) | , but, since X A 1 + A 2 is smo oth, we hav e: | det( a i − v , b k − b k − 1 ) | = ` ([ v , a i ]) · ` ([ b k − 1 , b k ]) = 1 · ` ([ b k − 1 , b k ]) Since E 0 2 ( v ) = { [ b 0 , b 1 ] , . . . , [ b s − 1 , b s ] } , this equality and Lemma 3.12 yield the result.  R emark 3.14 . The equality P v ∈ V ert A i mm( v ) = P e ∈E 0 j ` ( e ) in case i ( A 1 ) = i ( A 2 ) = 1 and the toric surfaces of A 1 , A 2 and A 1 + A 2 are smooth, can b e in terpreted and pro v ed with to ols form toric geometry . Indeed, in this case, let X 1 , X 2 and X b e the associated toric v arieties. Then, there are birational maps π i : X → X i defined by the common refinement of the asso ciated normal fans, i = 1 , 2. The map π 1 is giv en b y successiv e toric blow-ups of fixed p oints of X 1 corresp onding to v ertices v of A 1 for which E 0 2 ( v ) 6 = ∅ . The lengh ts of the corresp onding edges o ccur as the in tersection pro duct of the inv arian t (exceptional) divisor asso ciated to the edge with the ample line bundle asso ciated to A 2 , pulled back to X . If A i is dense, then it is immediate to c hec k that u ( e, A i ) = 1 for all edges e ≺ A i . W e conclude with a geometric upp er b ound for the bidegree of the mixed discriminan t. Corollary 3.15. L et A 1 and A 2 b e ful l-dimensional c onfigur ations in Z 2 . Then: (1) The bide gr e e satisfies deg A i ( ˜ ∆ A 1 ,A 2 ) 6 area( Q j ) + 2 MV ( Q 1 , Q 2 ) − p erim( Q j ) , j 6 = i . 12 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS (2) Equality holds in (1) if i ( A 1 ) = i ( A 2 ) = 1 and the thr e e toric surfac es of A 1 , A 2 and A 1 + A 2 ar e smo oth. (3) Equality holds in (1) if Q 1 , Q 2 have the same normal fan and one of A 1 or A 2 is dense. Pr o of. Assume i = 1. Statement (1) follo ws from ( 3.3 ) and X ( e,f ) ∈P min { u ( e, A 1 ) , u ( f , A 2 ) } ` ( f ) + X v ∈ A 1 mm( v ) > X f ∈E 2 \ E 0 2 ` ( f ) + X f ∈E 0 2 ` ( f ) = p erim( A 2 ) . Statemen t (2) follo ws from Theorem 3.13 (3) and the fact that the smoothness condition implies u ( e, A 1 ) = u ( f , A 2 ) = 1 for all edges e ≺ A 1 , f ≺ A 2 . Finally , if Q 1 and Q 2 ha ve the same normal fan then E 0 1 = E 0 2 = ∅ , and, by Theorem 3.13 (2), all mixed m ultiplicities v anish. Densit y of A 1 or A 2 implies min { u ( e, A 1 ) , u ( f , A 2 ) } = 1 for every pair ( e, f ) ∈ P . Hence X ( e,f ) ∈P min { u ( e, A 1 ) , u ( f , A 2 ) } ` ( f ) = p erim( Q 2 ) .  Corollary 3.15 establishes the degree form ula ( 1.3 ). W e end this section with an example for whic h that form ula holds, ev en though conditions (2) and (3) do not. It also shows that, unlik e for resultants [ 7 , § 6], the degree of the mixed discriminant can decrease when remo ving a single p oint from A without altering the lattice or the conv ex h ulls of the configurations. Example 3.16. Consider the dense configurations A 1 := { (0 , 0) , (1 , 0) , (1 , 1) , (0 , 1) } and A 2 := { (0 , 0) , (1 , 3) , ( − 1 , 2) , (0 , 1) , (0 , 2) } . The v ertex v = (0 , 0) of A 2 is a singular p oin t. Ho wev er, its mixed multiplicit y equals 1, so it agrees with the lattice length of the asso ciated edge [(0 , 0) , (1 , 0)] in A 1 . Theorem 3.3 implies that the bidegree of the mixed discriminant ∆ A 1 ,A 2 equals ( δ 1 , δ 2 ) = (12 , 8). If w e remov e the p oin t (0 , 1) from A 2 , the mixed multiplicit y of v is raised to 2 and the bidegree of the mixed discriminant decreases to (12 , 7). ♦ 4. The Degree of the Mixed Discriminant is Piecewise Linear Theorem 3.3 implies that the bidegree of the mixed discriminan t of A 1 , A 2 ⊂ Z 2 is piecewise linear in the maximal minors of the Ca yley matrix A = Cay( A 1 , A 2 ). In this section w e pro ve Theorem 1.1 which extends the same statement to arbitrary Cayley configurations, and we describ e suitable regions of linearity . Theorem 1.1 allo ws us to obtain form ulas for the m ultidegree of the mixed discriminant by linear algebraic metho ds, provided we are able to compute it in sufficien tly man y examples. This may b e done b y using the r ay sho oting algorithm of [ 7 , Theorem 2.2], which has b ecome a standard technique in tropical geometry . W e start by an example in dimension 3, whic h was computed using Rinc´ on’s softw are [ 16 ]. Example 4.1. Consider the following three trinomials in three v ariables: f = a 1 x + a 2 y p + a 3 z p , g = b 1 x q + b 2 y + b 3 z q , h = c 1 x r + c 2 y r + c 3 z . Here p, q and r are arbitrary integers different from 1. By the degree w e mean the triple of in tegers that records the degrees of the mixed discriminant cycle ∆( f , g , h ) in the unknowns MIXED DISCRIMINANTS 13 ( a 1 , a 2 , a 3 ), ( b 1 , b 2 , b 3 ), and ( c 1 , c 2 , c 3 ). It equals the following triple of piecewise p olynomials:  2 pq r + q 2 r + q r 2 − q − r − 1 − p min { q , r } , 2 pq r + p 2 r + pr 2 − p − r − 1 − q min { r , p } , 2 pq r + p 2 q + pq 2 − p − q − 1 − r min { p, q }  These three p olynomials are linear functions in the 6 × 6-minors of the 6 × 9 Cayley matrix A that represen ts ( f , g , h ). The space of all systems of three trinomials will b e defined as a certain mixe d Gr assmannian . The 6 × 6-minors represent its Pl ¨ uc ker co ordinates. ♦ Giv en m ∈ N with n 6 m , consider a partition I = { I 1 , . . . , I n } of the set [ m ] = { 1 , . . . , m } . Let G ( d, m ) denote the affine c one over the Gr assmannian of d -dimensional linear subspaces of R m , given b y its Pl ¨ uck er embedding in ∧ d R m . Thus G ( d, m ) is the subv ariet y of ∧ d R m cut out b y the quadratic Pl ¨ uc k er relations. F or instance, for d = 2 , m = 4, this is the hypersurface G (2 , 4) in ∧ 2 R 4 ' R 6 defined by the unique Pl ¨ uc ker relation x 12 x 34 − x 13 x 24 + x 14 x 23 = 0. Definition 4.2. The mixe d Gr assmannian G ( d, I ) asso ciated to the partition I is defined as the linear sub v ariet y of G ( d, m ) consisting of all subspaces that contain the v ectors e I j := P i ∈ I j e i for j = 1 , . . . , n . Here “linear subv ariet y” means that G ( d, I ) is the intersection of G ( d, m ) with a linear subspace of the  m d  -dimensional real vector space ∧ d R m . The condition that a subspace ξ contains e I j translates in to a system of n ( m − d ) linearly indep enden t linear forms in the Pl¨ uck er co ordinates that v anish on G ( d, I ). These linear forms are obtained as the co ordinates of the exterior pro ducts ξ ∧ e I j for j = 1 , . . . , n . W e should stress one crucial p oin t. As an abstract v ariety , the mixed Grassmannian G ( d, I ) is isomorphic to the ordinary Grassmannian G ( d − n, m − n ), where the isomorphism maps ξ to its image mo dulo span( e I 1 , . . . , e I n ). Ho w ever, we alwa ys work with the Pl ¨ uck er co ordinates of the ambien t Grassmannian G ( d, m ) in ∧ d R m . W e do not consider the mixed Grassmannian G ( d − n, m − n ) in its Pl ¨ uc ker embedding in ∧ d − n R m − n . Our mixed Grassmannian has a natural decomp osition in to finitely many strata whose definition in volv es oriente d matr oids . On eac h stratum, the degree of the mixed discriminan t cycle is a linear function in the Pl¨ uc k er coordinates. In order to define tropical matroid strata and to prov e Theorem 1.1 , it will b e conv enien t to regard the mixed discriminant as the A - discriminan t ∆ A of the Ca yley matrix A . In fact, w e shall consider ∆ A for arbitrary matrices A ∈ Z d × m of rank d such that e [ m ] = (1 , 1 , . . . , 1) is in the ro w span of A . Then, A represents a p oin t ξ in the Grassmannian G ( d, { [ m ] } ). This is the proper subv ariety of G ( d, m ) consisting of all p oints whose subspace contains e [ m ] . In what follo ws w e assume some familiarit y with matroid theory and tropical geometry . W e refer to [ 7 , 11 ] for details. Given a d × m -matrix A of rank d as ab o ve, we let M ∗ ( A ) denote the corresp onding dual matroid on [ m ]. This matroid has rank m − d . A subset I = { i 1 , . . . , i r } ⊆ [ m ] is indep endent in M ∗ ( A ) if and only if e ∗ i 1 , . . . , e ∗ i r are linearly independent when restricted to k er( A ), where e ∗ 1 , . . . , e ∗ n denotes the standard dual basis. The flats of the matroid M ∗ ( A ) are the subsets J ⊆ [ m ] such that [ m ] \ J is the supp ort of a vector in ker( A ). Let T (ker( A )) denote the tropicalization of the kernel of A . This tropical linear space is a balanced fan of dimension m − d in R m . It is also known as the Ber gman fan of M ∗ ( A ), and it admits v arious fan structures [ 11 , 16 ]. Ardila and Kliv ans [ 1 ] show ed that the chains in the geometric lattice of M ∗ ( A ) endow the tropical linear space T (ker( A )) with the structure of a simplicial fan. The cones in this fan are span( e J 1 , e J 2 , . . . , e J r ) where 14 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS J = { J 1 ⊂ J 2 ⊂ · · · ⊂ J r } runs ov er all chains of flats of M ∗ ( A ). Such a cone is maximal when r = m − d − 1. Given any suc h maximal chain and any index i ∈ [ m ], w e asso ciate with them the follo wing m × m matrix: M ( A, J , i ) := ( A T , e J 1 , e J 2 , . . . , e J m − d − 1 , e i ) . Its determinant is a linear expression in the Pl ¨ uck er co ordinates of the ro w span ξ of A : det( M ( A, J , i )) = ξ ∧ e J 1 ∧ e J 2 ∧ · · · ∧ e J m − d − 1 ∧ e i . Definition 4.3. Let A and A 0 b e matrices represen ting p oin ts ξ and ξ 0 in G ( d, { [ m ] } ). These p oin ts b elong to the same tr opic al matr oid str atum if they hav e the same dual matroid, i.e., M ∗ ( A ) = M ∗ ( A 0 ) , and, in addition, for all i ∈ [ m ] and all maximal chains of flats J in the ab o v e matroid, the determinan ts of the matrices M ( A, J , i ) and M ( A 0 , J , i ) hav e the same sign. R emark 4.4 . Dick enstein et al. [ 7 ] gav e the follo wing form ula for the tr opic al A -discriminant : (4.1) T (∆ A ) = T (k er( A )) + ro wspan( A ) . This is a tropical cycle in R m , i.e. a polyhedral fan that is balanced relativ e to the m ultiplic- ities asso ciated to its maximal cones. The dimension of T (∆ A ) equals m − 1 whenev er A is not defectiv e. It is clear from the form ula ( 4.1 ) that T (∆ A ) dep ends only on the subspace ξ = ro wspan( A ), so it is a function of ξ ∈ G ( d, { [ m ] } ). The tropical matroid strata are the subsets of G ( d, { [ m ] } ) throughout which the com binatorial type of ( 4.1 ) do es not change. Example 4.5. W e illustrate the definition of the tropical matroid strata by revisiting the form ulas in ( 1.6 ) and Example 3.10 . The Ca yley matrix of the tw o sparse triangles equals A = Ca y( A 1 , A 2 ) =     1 1 1 0 0 0 0 0 0 1 1 1 0 d 1 0 0 d 2 0 0 0 d 1 0 0 d 2     . The matroid M ∗ ( A ) has rank 2, so every maximal c hain of flats in M ∗ ( A ) consists of a single rank 1 flat. These flats are J 1 = { 1 , 4 } , J 2 = { 2 , 5 } , and J 3 = { 3 , 6 } . The 6 × 6-determinan ts det( M ( A, J , i )) obtained b y augmenting A with one vector e J k and one unit vector e i are 0, ± d 1 ( d 1 − d 2 ), or ± d 2 ( d 1 − d 2 ). This shows that d 1 > d 2 > 0 and d 1 > 0 > d 2 are tropical matroid strata, corresp onding to ( 1.6 ) and to Example 3.10 with d 2 replaced by − d 2 . ♦ R emark 4.6 . The verification that tw o configurations lie in the same tropical matroid stratum ma y inv olve a huge num b er of maximal flags if w e use Definition 4.3 as it is. In practice, we can greatly reduce the num b er of signs of determinan ts to b e chec k ed, b y utilizing a coarser fan structure on T (k er( A )). The coarsest fan structure is given by the irr e ducible flats and their neste d sets , as explained in [ 11 ]. Rather than reviewing these com binatorial details for arbitrary matrices, we simply illustrate the resulting reduction in complexity when M ∗ ( A ) is the uniform matr oid . This means that an y d columns of A form a basis of R d . Then, M ∗ ( A ) has ( m − d − 1)!  m m − d − 1  maximal flags J 1 ⊂ J 2 ⊂ · · · ⊂ J m − d − 1 constructed as follows. Let I = { i 1 , . . . , i m − d − 1 b e an ( m − d − 1)-subset of [ m ] and σ a permutation of [ m − d − 1]. Then, w e set J k := [ m ] \ { i σ (1) , . . . , i σ ( k ) } . It is clear that the sign of det( A T , e J 1 , e J 2 , . . . , e J m − d − 1 , e k ) is completely determined b y the signs of the determinants det( A T , e i 1 , e i 2 , . . . , e i m − d − 1 , e k ) , MIXED DISCRIMINANTS 15 where i 1 < i 2 < · · · < i m − d − 1 . Hence, we only need to chec k  m m − d − 1  conditions. Recall that the A -discriminant cycle ˜ ∆ A = ∆ i ( A ) A is effective of co dimension 1, provided A is non-defective. The lattice index i ( A ) is the gcd of all maximal minors of A . Theorem 4.7. The de gr e e of the A -discriminant cycle is pie c ewise line ar in the Pl ¨ ucker c o or dinates on G ( d, { [ m ] } ) . It is line ar on the tr opic al matr oid str ata. The formulas on maximal str ata ar e unique mo dulo the line ar forms obtaine d fr om the entries of ξ ∧ e [ m ] . In b oth Theorem 1.1 and Theorem 4.7 , the notion of “degree” allows for an y grading that mak es the resp ectiv e discriminan t homogeneous. F or the mixed discriminan t ∆ A 1 ,...,A n w e are in terested in the N n -degree. Theorem 1.1 will b e deriv ed as a corollary from Theorem 4.7 . Pr o of of The or em 4.7 . The uniqueness of the degree form ula follo ws from our earlier remark that the en tries of ξ ∧ e [ m ] are the linear relations on the mixed Grassmannian G ( d, { [ m ] } ). W e no w show ho w tropical geometry leads to the desired piecewise linear formula. F rom the represen tation of the tropical discriminant in ( 4.1 ), Dic k enstein et al. [ 7 , Theorem 5.2] deriv ed the follo wing form ula for the initial monomial of the A -discriminant ∆ A with resp ect to any generic weigh t vector ω ∈ R m . The exp onent of the v ariable x i in the initial monomial in ω (∆ A ) of the A -discriminan t ∆ A is equal to (4.2) X J ∈C i,ω | det( A T , e J 1 , . . . , e J m − d − 1 , e i ) | . Here, C i,ω is the set of maximal c hains J of M ∗ ( A ) suc h that the ro wspan ξ of A has non-zero in tersection with the relativ ely op en cone R > 0  e J 1 , . . . , e J m − d − 1 , − e i , − ω  . It now suffices to pro v e the following statement: if t w o matrices A and A 0 lie in the same tropical matroid stratum, then there exists weigh t v ectors ω and ω 0 suc h that C i,ω = C i,ω 0 . This ensures that the sum in ( 4.2 ), with the absolute v alue replaced with the appropriate sign, yields a linear function in the Pl ¨ uc ker co ordinates of ξ for the degree of ∆ A and ∆ A 0 . The condition J ∈ C i,ω is equiv alent to the weigh t vector ω b eing in the cone R > 0  e J 1 , . . . , e J m − d − 1 , − e i  + ξ . Hence, it is con v enient to w ork mo dulo ξ . This amounts to considering the exact sequence 0 − − − → R d A T − − − → R k β − − − → W − − − → 0 . Cho osing a basis for k er( A ), we can iden tify W ' R m − d . The columns of the matrix β define a vector configuration B = { b 1 , . . . , b m } ⊂ R m − d called a Gale dual c onfigur ation of A . Pro jecting in to W , w e see that C i,ω equals the set of all maximal c hains J such that β ( ω ) lies on the cone R > 0  β ( e J 1 ) , . . . , β ( e J m − d − 1 ) , − β ( e i )  . It follows that J ∈ C i,ω if and only if β ( ω ) = m X j =1 w j b j ∈ R > 0  σ J 1 , . . . , σ J m − d − 1 , − b i  , where σ J := β ( e J ) = P j ∈J b j . W e can also restate the definition of the tropical matroid strata in terms of Gale duals. Namely , there exists a non-zero constant c , dep ending only on d , m and our c hoice of Gale dual B , such that, giv en a maximal chain of flats J in the matroid M ∗ ( A ), we hav e: (4.3) det( A T , e J 1 , e J 2 , . . . , e J m − d − 1 , e i ) = c · det( σ J 1 , σ J 2 , . . . , σ J m − d − 1 , b i ) . 16 E. CA TT ANI, M. A. CUETO, A. DICKENSTEIN, S. DI ROCCO, AND B. STURMFELS Hence, the tropical matroid strata in G ( d, { [ m ] } ) are determined by the signs of the deter- minan t on the right-hand side of ( 4.3 ). If J ∈ C i,ω for generic ω ∈ R m , then the vectors { σ J 1 , . . . , σ J m − d − 1 , b i } in R m − d are linearly indep enden t. Let M ( J , B , i ) b e the matrix whose columns are these v ectors. Then, J ∈ C i,ω if and only if the vector x = M ( J , B , i ) − 1 β ( ω ) has p ositiv e entries. By Cramer’s rule, those en tries are (4.4)        x k = det( σ J 1 , . . . , σ J k − 1 , β ( ω ) , σ J k +1 , . . . , σ J m − d − 1 , − b i ) det( M ( J , B , i )) for 0 6 k < m − d, x m = det( σ J 1 , σ J 2 , . . . , σ J m − d − 1 , β ( ω )) det( M ( J , B , i )) . Supp ose no w that A and A 0 are t wo configurations in the same tropical matroid stra- tum. Let B and B 0 b e their Gale duals. Then M ∗ ( A ) = M ∗ ( A 0 ) and the denominators det( M ( J , B , i )) and det( M ( J , B 0 , i )) in ( 4.4 ) hav e the same signs. On the other hand, let us consider the oriented hyperplane arrangemen t in R m − d consisting of the h yp erplanes H J ,B ,k ,i = h σ J 1 , . . . , σ J k − 1 , σ J k +1 , . . . , σ J m − d − 1 , b i i , for 1 6 k 6 m − d − 1 , i 6∈ J m − d − 1 , as w ell as the h yp erplane H J = h σ J 1 , . . . , σ J m − d − 1 i , for all maximal c hains J ∈ M ∗ ( A ) suc h that σ J 1 , . . . , σ J m − d − 1 are linearly indep endent. The signs of the n umerators in ( 4.4 ) are deter- mined by the orien ted h yp erplane arrangemen t just defined. Since M ∗ ( A ) = M ∗ ( A 0 ), w e can establish a corresp ondence b et ween the cells of the complemen ts of these arrangements that preserv es the signs in ( 4.4 ) for b oth A and A 0 , giv en w eights ω and ω 0 in corresp onding cells. This means that C i,ω = C i,ω 0 as we wan ted to show.  W e note that the conclusion of Theorem 4.7 is also v alid on tropical matroid strata where A is defectiv e. In that case the A -discriminant ∆ A equals 1, and its degree is the zero vector. W e end this section b y showing how to obtain our main result on mixed discriminants. Pr o of of The or em 1.1 . Suppose that A is the Ca yley matrix of n configurations A 1 , . . . , A n and let I = { I 1 , . . . , I n } b e the asso ciated partition of [ m ]. It follows from ( 4.2 ) that deg A k (∆ A ) = X i ∈ I k X J ∈C i,ω | det( A T , e J 1 , . . . , e J m − d − 1 , e i ) | . By the same argumen t as in the pro of of Theorem 4.7 , we conclude that the ab o v e expression defines a fixed linear form on ∧ d R m for all matrices A in a fixed tropical matroid stratum.  In closing, w e wish to reiterate that com bining Theorem 1.1 with Rinc´ on’s results in [ 16 ] leads to p o w erful algorithms for computing piecewise p olynomial degree form ulas. Here is an example that illustrates this. W e consider the n -dimensional version of the system ( 1.5 ): f i = c i 0 + c i 1 x d i 1 + c i 2 x d i 2 + · · · + c in x d i n for i = 1 , 2 , . . . , n, where 0 6 d 1 6 d 2 6 · · · 6 d n are coprime in tegers. The Ca yley matrix A has 2 n rows and n 2 + n columns. Using his softw are, F elip e Rinc´ on computed the corresp onding tropical discriminan t for n = 4, while keeping the d i as unknowns, and he found deg A i (∆ A 1 ,...,A n ) = d 1 · · · d i − 1 d i +1 · · · d n ·  d i + ( − n ) d 1 + d 2 + d 3 + · · · + d n  . Th us, w e ha ve a computational pro of of this form ula for n 6 4, and it remains a conjecture for n > 5. This sho ws ho w the findings of this section may b e used in experimental mathematics. MIXED DISCRIMINANTS 17 Ac knowledgmen ts: MA C w as supp orted b y an AXA Mittag-Leffler p ostdo ctoral fellowship (Sw eden) and an NSF p ostdo ctoral fellowship DMS-1103857 (USA). AD w as supp orted b y UBA CYT 20020100100242, CONICET PIP 112-200801-00483 and ANPCyT 2008-0902 (Argen tina). SDR was partially supp orted b y VR gran t NT:2010-5563 (Sweden). BS w as supp orted by NSF grants DMS-0757207 and DMS-0968882 (USA). This pro ject started at the Institut Mittag-Leffler during the Spring 2011 program on “Algebraic Geometry with a View T o w ards Applications.” W e thank IML for its wonderful hospitalit y . References [1] F. Ardila and C. J. Kliv ans. The Bergman complex of a matroid and phylogenetic trees. J. Combin. The ory Ser. B , 96 (1):38–49, 2006. [2] O. 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