Semi-inverted linear spaces and an analogue of the broken circuit complex
SSEMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THEBROKEN CIRCUIT COMPLEX
GEORGY SCHOLTEN AND CYNTHIA VINZANT
Abstract.
The image of a linear space under inversion of some coordinates is an affinevariety whose structure is governed by an underlying hyperplane arrangement. In thispaper, we generalize work by Proudfoot and Speyer to show that circuit polynomials forma universal Gröbner basis for the ideal of polynomials vanishing on this variety. The proofrelies on degenerations to the Stanley-Reisner ideal of a simplicial complex determined bythe underlying matroid, which is closely related to the external activity complex defined byArdila and Boocher. If the linear space is real, then the semi-inverted linear space is alsoan example of a hyperbolic variety, meaning that all of its intersection points with a largefamily of linear spaces are real. Introduction
In 2006, Proudfoot and Speyer showed that the coordinate ring of a reciprocal linearspace (i.e. the closure of the image of a linear space under coordinate-wise inversion) has aflat degeneration into the Stanley-Reisner ring of the broken circuit complex of a matroid[12]. This completely characterizes the combinatorial data of these important varieties,which appear across many areas of mathematics, including in the study of matroids andhyperplane arrangements [18], interior point methods for linear programming [4], and entropymaximization for log-linear models in statistics [9].In this paper we extend the results of Proudfoot and Speyer to the image of a linear space
L ⊂ C n under inversion of some subset of coordinates. For I ⊆ { , . . . , n } , consider therational map inv I : C n (cid:57)(cid:57)(cid:75) C n defined by (inv I ( x )) i = (cid:40) /x i if i ∈ Ix i if i (cid:54)∈ I. Let inv I ( L ) denote the Zariski-closure of the image of L under this map, which is an affinevariety in C n . One can interpret inv I ( L ) as an affine chart of the closure of L in the productof projective spaces ( P ) n , as studied in [1], or as the projection of the graph of L underthe map x (cid:55) (cid:57)(cid:57)(cid:75) inv [ n ] ( x ) , studied in [5], onto complementary subsets of the n coordinates.We give a degeneration of the coordinate ring of inv I ( L ) to the Stanley-Reisner ring ofa simplicial complex generalizing the broken circuit complex of a matroid. This involvesconstructing a universal Gröbner basis for the ideal of polynomials vanishing on inv I ( L ) .Let C [ x ] denote the polynomial ring C [ x , . . . , x n ] and for any α ∈ ( Z ≥ ) n , let x α denote (cid:81) ni =1 x α i i . For a subset S ⊆ [ n ] , we will also use x S to denote (cid:81) i ∈ S x i . As in [12], the circuits of the matroid M ( L ) corresponding to L give rise to a universal Gröbner basis for the idealof polynomials vanishing on inv I ( L ) . We say that a linear form (cid:96) ( x ) = (cid:80) i ∈ [ n ] a i x i vanisheson L if (cid:96) ( x ) = 0 for all x ∈ L . The support of (cid:96) , supp( (cid:96) ) , is { i ∈ [ n ] : a i (cid:54) = 0 } . The minimal Date : June 10, 2019. a r X i v : . [ m a t h . C O ] J un GEORGY SCHOLTEN AND CYNTHIA VINZANT supports of nonzero linear forms vanishing on L are called circuits of the matroid M ( L ) and for every circuit C ⊂ [ n ] , there is a unique (up to scaling) linear form (cid:96) C = (cid:80) i ∈ C a i x i vanishing on L with support C . To each circuit, we associate the polynomial(1) f C ( x ) = x C ∩ I · (cid:96) C (inv I ( x )) = (cid:88) i ∈ C ∩ I a i x C ∩ I \{ i } + (cid:88) i ∈ C \ I a i x C ∩ I ∪{ i } . Theorem 1.1.
Let
L ⊆ C n be a d -dimensional linear space and let I ⊆ C [ x ] be the idealof polynomials vanishing on inv I ( L ) . Then { f C : C is a circuit of M ( L ) } is a universalGröbner basis for I . For w ∈ ( R + ) n with distinct coordinates, the initial ideal in w ( I ) is theStanley-Reisner ideal of the semi-broken circuit complex ∆ w ( M ( L ) , I ) . The simplicial complex ∆ w ( M ( L ) , I ) will be defined in Section 3. For real linear spaces L , the variety inv I ( L ) relates to the regions of a hyperplane arrangement. Theorem 1.2.
Let
L ⊆ C n be a linear space that is invariant under complex conjugation.Then the following numbers are equal: (1) the degree of the affine variety inv I ( L ) , (2) the number of facets of the semi-broken circuit complex ∆ w ( M ( L ) , I ) , and (3) for generic u ∈ R n , the number of regions in ( L ⊥ + u ) \{ x i = 0 } i ∈ I whose recessioncones trivially intersect R I = { x ∈ R n : x j = 0 for j (cid:54)∈ I } . The paper is organized as follows. The necessary definitions and background on matroidtheory and Stanley-Reisner ideals are in Section 2. In Section 3 we define the simplicialcomplex ∆ w ( M ( L ) , I ) , show that it satisfies a deletion-contraction relation analogous tothat of the broken circuit complex of a matroid, and describe its relationship to the externalactivity complex of a matroid. Section 4 contains the proof of Theorem 1.1. We characterizethe strata of inv I ( L ) given by its intersection with coordinate subspaces in Section 5. Finally,in Section 6, we show that for a real linear space L , inv I ( L ) is a hyperbolic variety , in thesense of [7, 15], and prove Theorem 1.2. Acknowledgements.
The authors would like to thank Nicholas Proudfoot, Seth Sul-livant, and Levent Tunçel for useful discussions over the course of this project and theanonymous referees for their careful reading and helpful suggestions.2.
Background
In this section, we review the necessary background on Gröbner bases, simplicial com-plexes, Stanley-Reisner ideals, matroids, and previous research on reciprocal linear spaces.2.1.
Gröbner bases and degenerations.
A finite subset F of an ideal I ⊂ C [ x ] is a universal Gröbner basis for I if it is a Gröbner basis with respect to every monomial orderon C [ x ] . An equivalent definition using weight vectors is given as follows. For w ∈ ( R ≥ ) n and f = (cid:80) α c α x α ∈ C [ x ] , define the degree and initial form of f with respect to w to be deg w ( f ) = max { w T α : c α (cid:54) = 0 } and in w ( f ) = (cid:88) α : w T α =deg w ( f ) c α x α . The initial ideal in w ( I ) of an ideal I is the ideal generated by initial forms of polynomials in I , i.e. in w ( I ) = (cid:104) in w ( f ) : f ∈ I(cid:105) . Then F ⊂ I is a universal Gröbner basis for I if and onlyif for every w ∈ ( R ≥ ) n , the polynomials in w ( F ) generate in w ( I ) . i.e. See [17, Chapter 1]. EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 3
For homogeneous I , in w ( I ) is a flat degeneration of I . For f ∈ C [ x ] and an integer vector w ∈ ( Z ≥ ) n , define t w · f = t deg w ( f ) f ( t − w x , . . . , t − w n x n ) ∈ C [ t, x ] and I w = (cid:104) t w · f : f ∈ I(cid:105) ⊂ C [ t, x ] . The ideal I w defines a variety in A ( C ) × P n − ( C ) , namely the Zariski-closure V ( I w ) = (cid:8) ( t, [ t w x : . . . : t w n x n ]) such that t ∈ C ∗ , x ∈ V ( I ) (cid:9) Zar . Letting t vary from to gives a flat deformation from V ( I ) to the variety V (in w ( I )) .Formally, for any γ ∈ C , let I w ( γ ) denote the ideal in C [ x ] obtained by substituting t = γ .Then I w (1) equals I , I w (0) equals in w ( I ) , and for γ ∈ C ∗ , the variety of I w ( γ ) consists ofthe points { [ γ w x : . . . : γ w n x n ] : x ∈ V ( I ) } . We note that the ideal I w ( γ ) is well-definedfor any w ∈ R n . All the ideals I w ( γ ) have the same Hilbert series. In particular, taking γ = 0 , shows that I and in w ( I ) have the same Hilbert series.2.2. Simplicial complexes and Stanley-Reisner ideals.
A Stanley-Reisner ideal is asquare-free monomial ideal. Its combinatorial properties are governed by a simplicial com-plex. A simplicial complex ∆ on vertices { , . . . , n } is a collection of subsets of { , . . . , n } ,called faces, that is closed under taking subsets. If S ∈ ∆ has cardinality k + 1 , we call it aface of dimension k . A facet of ∆ is a face maximal in ∆ under inclusion. Given a simplicialcomplex ∆ on [ n ] \{ i } , define the cone of ∆ over i to be cone(∆ , i ) = ∆ ∪ { S ∪ { i } : S ∈ ∆ } , which is a simplicial complex on [ n ] whose facets are in bijection with the facets of ∆ . Definition 2.1 (See e.g. [16, Chapter II]) . Let ∆ be a simplicial complex on vertices { , . . . , n } . The Stanley-Reisner ideal of ∆ is the square-free monomial ideal I ∆ = (cid:10) x S : S ⊆ [ n ] , S (cid:54)∈ ∆ (cid:11) generated by monomials corresponding to the non-faces of ∆ . The Stanley-Reisner ring of ∆ is the quotient ring C [ x ] / I ∆ .The ideal I ∆ is radical and it equals the intersection of prime ideals I ∆ = (cid:92) F a facet of ∆ (cid:104) x i : i (cid:54)∈ F (cid:105) . This writes the variety V ( I ∆ ) as the union of coordinate subspaces span { e i : i ∈ F } where F is a facet of ∆ . In particular, if ∆ has k facets of dimension d − , then V ( I ∆ ) ⊆ P n − ( C ) is a variety of dimension d − and degree k . See [16, Chapter II].2.3. Matroids.
Matroids are a combinatorial model for many types of independence rela-tions. See [11] for general background on matroid theory. We can associate a matroid M ( L ) to a linear space L ⊂ C n as follows. Write a d -dimensional linear space L ⊂ C n as therowspan of a d × n matrix A = ( a , . . . , a n ) . A set I ⊆ [ n ] is independent in M ( L ) if thevectors { a i : i ∈ I } are linearly independent in C d . For any invertible matrix U ∈ C d × d ,the vectors { a i : i ∈ I } are linearly independent if and only if the vectors { U a i : i ∈ I } arealso independent, implying that this condition is independent of the choice of basis for L .Indeed, I ⊆ [ n ] is independent in M ( L ) if and only if the coordinate linear forms { x i : i ∈ I } are linearly independent when restricted to L . GEORGY SCHOLTEN AND CYNTHIA VINZANT
Maximal independent sets are called bases and minimal dependent sets are called cir-cuits . We use B ( M ) and C ( M ) to denote the set of bases and circuits of a matroid M ,respectively. An element i ∈ [ n ] is called a loop if { i } is a circuit, and a co-loop if i is contained in every basis of M . The rank of a subset S ⊆ [ n ] is the largest size of anindependent set in S . A flat is a set F ⊆ [ n ] that is maximal for its rank, meaning that rank( F ) < rank( F ∪ { i } ) for any i (cid:54)∈ F .Let M be a matroid on [ n ] and i ∈ [ n ] . The deletion of M by i , denoted M \ i , isthe matroid on the ground set [ n ] \ i whose independent sets are subsets I ⊂ [ n ] \ i that areindependent in M . If i is not a co-loop of M , then B ( M \ i ) = { B ∈ B ( M ) : i / ∈ B } and C ( M \ i ) = { C ∈ C ( M ) : i / ∈ C } . More generally, the deletion of M by a subset S ⊂ [ n ] , denoted M \ S , is the matroid obtainedfrom M by successive deletion of the elements of S . The restriction of M to a subset S ,denoted M | S , is the deletion of M by [ n ] \ S .If i is not a loop of M , then the contraction of M by i , denoted M/i , is the matroidon the ground set [ n ] \{ i } whose independent sets are subsets I ⊂ [ n ] \ i for which I ∪ { i } isindependent in M . Then B ( M/i ) = { B \ i : B ∈ B ( M ) , i ∈ B } , and C ( M/i ) = inclusion minimal elements of { C \ i : C ∈ C ( M ) } . If i is a loop of M , then we define the contraction of M/i to be the deletion M \ i . Thecontraction of M by a subset S ⊂ [ n ] , denoted M/S , is obtained from M by successivecontractions by the elements of S .For linear matroids, deletion and contraction correspond to projection and intersection inthe following sense. For S ⊂ [ n ] , let L\ S denote the linear subspace of C [ n ] \ S obtained byprojecting L away from the coordinate space C S = span { e i : i ∈ S } . Let L /S denote theintersection of L with C [ n ] \ S . Then M ( L ) \ S = M ( L\ S ) and M ( L ) /S = M ( L /S ) . Many interesting combinatorial properties of a matroid can be extracted from a simplicialcomplex called the broken circuit complex. Given a matroid M and the usual ordering < < . . . < n on [ n ] , a broken-circuit of M is a subset of the form C \ min( C ) where C ∈ C ( M ) . The broken-circuit complex of M is the simplicial complex on [ n ] whose facesare the subsets of [ n ] not containing any broken circuit.2.4. Reciprocal linear spaces.
For I = [ n ] , the variety inv I ( L ) is well-studied in theliterature. Proudfoot and Speyer study the coordinate ring of the variety inv [ n ] ( L ) and relateit to the broken circuit complex of a matroid [12]. One of their motivations is connectionswith the cohomology of the complement of a hyperplane arrangement. These varieties alsoappear in the algebraic study of interior point methods for linear programming [4] andentropy maximization for log-linear models in statistics [9].If the linear space L is invariant under complex conjugation, the variety inv [ n ] ( L ) also hasa special real-rootedness property. Specifically, if L ⊥ denotes the orthogonal complement of L , then for any u ∈ R n , all the intersection points of inv [ n ] ( L ) and the affine space L ⊥ + u are real. This was first shown in different language by Varchenko [19] and used extensivelyin [4]. One implication of this real-rootedness is that the discriminant of the projectionaway from L ⊥ is a nonnegative polynomial [14]. Another is that inv [ n ] ( L ) is a hyperbolic EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 5 variety, in the sense of [15]. In fact, the Chow form of the variety inv [ n ] ( L ) has a definitedeterminantal representation, certifying its hyperbolicity [7]. We generalize some of theseresults to inv I ( L ) .In closely related work [1], Ardila and Boocher study the closure of a linear space L inside of ( P ) n . For any I ⊆ [ n ] , inv I ( L ) can be considered as an affine chart of thisprojective closure. Specifically, let X ⊆ ( P ) n denote the closure of the image of L under ( x , . . . , x n ) (cid:55)→ ([ x : y ] , [ x : y ] , . . . , [ x n : y n ]) , with y = . . . = y n = 1 . The restriction of X to the affine chart x i = 1 for i ∈ I and y j = 1 for j ∈ [ n ] \ I is isomorphic to inv I ( L ) . Thecohomology and intersection cohomology of the projective variety X have been studied togreat effect in [6] and [13]. The precise relationship between the external activity complex ofa matroid used in [1] and the semi-broken circuit complex is described at the end of Section 3.3. A semi-broken circuit complex
Let M be a matroid on elements [ n ] and suppose I ⊆ [ n ] . A vector w ∈ R n with distinctcoordinates gives an ordering on [ n ] , where i < j whenever w i < w j . Without loss ofgenerality, we can assume w < . . . < w n , which induces the usual order < . . . < n . Givena circuit C of M we define an I -broken circuit of M to be b I ( C ) = (cid:40) C \ min( C ) if C ⊆ I ( C ∩ I ) ∪ max( C \ I ) if C (cid:54)⊆ I. Now we define the I -broken circuit complex of M to be(2) ∆ w ( M, I ) = { τ ⊆ [ n ] : τ does not contain an I -broken circuit of M } . Note that an [ n ] -broken circuit is a broken circuit in the usual sense and ∆ w ( M, [ n ]) is thewell-studied broken circuit complex of M . Example 3.1.
Consider the rank- matroid on [5] with circuits C = { , , } . Let I = { , , } and suppose w ∈ ( R + ) with w < . . . < w . Then its I -broken circuits are b I (124) = 124 , b I (135) = 135 , and b I (2345) = 235 . The simplicial complex ∆ w ( M, I ) is apure -dimensional simplicial complex with facets: facets(∆ w ( M, I )) = { , , , , , , } . The I -broken circuit complex shares many properties with the classical one, which willimply that it is always pure of dimension rank( M ) − . Theorem 3.2.
Let ∆ w ( M, I ) be the I -broken circuit complex defined in (2) . (a) If i ∈ I is a loop of M , then ∆ w ( M, I ) = ∅ . (b) If i ∈ I is a coloop of M , then ∆ w ( M, I ) = cone(∆ w ( M/i, I \ i ) , i ) . (c) If i = max( I ) is neither a loop nor a coloop of M , then ∆ w ( M, I ) = ∆ w ( M \ i, I \ i ) ∪ cone(∆ w ( M/i, I \ i ) , i ) . Proof. (a) If i ∈ I is a loop, then C = { i } is a circuit of M with b I ( C ) = ∅ .(b) If i ∈ I is a coloop, then no circuit of M , and hence no I -broken circuit, contains i .The circuits of M are exactly the circuits of the contraction M/i and the I -broken circuitsof M are the ( I \ i ) -broken circuits of M/i . Therefore τ is a face of ∆ w ( M, I ) if any only if τ \ i is a face of ∆ w ( M/i, I \ i ) .(c) ( ⊆ ) Let τ be a face of ∆ w ( M, I ) . We will show that if i (cid:54)∈ τ , then τ is a face of ∆ w ( M \ i, I \ i ) and if i ∈ τ , then τ \ i is a face of ∆ w ( M/i, I \ i ) . GEORGY SCHOLTEN AND CYNTHIA VINZANT If i (cid:54)∈ τ and C is a circuit of the deletion M \ i , then C is a circuit of M , and b I ( C ) = b I \ i ( C ) is an I -broken circuit of M and therefore is not contained in τ . If i ∈ τ and C is a circuit ofthe contraction M/i , then either C or C ∪{ i } is a circuit of M . In the first case, we again havethat b I ( C ) = b I \ i ( C ) is not contained in τ and thus not contained in τ \ i . Secondly, supposethat C ∪ { i } is a circuit of M . If C ⊆ I , then b I ( C ∪ { i } ) is equal to C ∪ { i }\ min( C ∪ { i } ) .Since i is the maximum element of I , this equals C \ min( C ) ∪ { i } . This set is not containedin τ . Therefore b I \ i ( C ) = C \ min( C ) is not contained in τ \ i . If C (cid:54)⊆ I , then the I -brokencircuit of C ∪ { i } is ( C ∩ I ) ∪ { i } ∪ max( C \ I ) , which equals b I \ i ( C ) ∪ { i } . Since τ cannotcontain an I -broken circuit of M , τ \ i does not contain b I \ i ( C ) .( ⊇ ) Let τ be a face of ∆ w ( M \ i, I \ i ) and suppose C is a circuit of M . If i (cid:54)∈ C , then C isalso a circuit of M \ i , implying that b I ( C ) is not contained in τ . If i ∈ C and C ⊆ I , then i = max( C ) . Since i is not a loop, this implies that i ∈ b I ( C ) , which cannot be contained in τ . Similarly, if i ∈ C and C (cid:54)⊂ I , then i ∈ b I ( C ) and b I ( C ) (cid:54)⊂ τ .Finally, let τ be a face of ∆ w ( M/i, I \ i ) and let C be a circuit of M . If i ∈ C , then C \ i isa circuit of M/i . Then b I ( C ) equals b I \ i ( C \ i ) ∪ { i } . Since τ cannot contain b I \ i ( C \ i ) , τ ∪ { i } does not contain b I ( C ) . If i (cid:54)∈ C , then C is a union of circuits of M/i , see [11, §3.1, Exercise2]. If C ⊆ I , then there is a circuit C (cid:48) ⊆ C of M/i containing min( C ) . Then C (cid:48) ⊆ I \ i and b I \ i ( C (cid:48) ) is a subset of b I ( C ) . Similarly, if C (cid:54)⊆ I , then there is a circuit C (cid:48) ⊆ C of M/i containing max( C \ I ) , giving b I \ i ( C (cid:48) ) ⊆ b I ( C ) . In either case, τ is a face of ∆ w ( M/i, I \ i ) andcannot contain the broken circuit b I \ i ( C (cid:48) ) and therefore τ ∪ { i } cannot contain b I ( C ) . (cid:3) Corollary 3.3. If M is a matroid of rank d with no loops in I , then ∆ w ( M, I ) is a puresimplicial complex of dimension d − .Proof. We induct on the size of I . If I = ∅ , then for every circuit C , the broken circuit b I ( C ) is the maximum element max( C ) . In this case, the simplicial complex ∆ w ( M, I ) consistsof one maximal face B , where B is the lexicographically smallest basis of M ( L ) . Here B consists of the elements i ∈ [ n ] for which the rank of [ i ] in M ( L ) is strictly larger than therank of [ i − . Every other element is the maximal element of some circuit of M .Now suppose | I | > and consider i = max( I ) . If i is a coloop of M , then the contraction M/i is a matroid of rank d − with no loops in I \ i . Then by induction and Theorem 3.2(b), ∆ w ( M, I ) = cone(∆ w ( M/i, I \ i ) , i ) is a pure simplicial complex of dimension d − . Finally,suppose i is neither a loop nor a coloop of M . Then the deletion M \ i is a matroid of rank d and no element of I \ i is a loop of M \ i . It follows that ∆ w ( M \ i, I \ i ) is a pure simplicialcomplex of dimension d − . The contraction M/i is a matroid of rank d − , implying that ∆ w ( M/i, I \ i ) is either empty (if I \ i contains a loop of M/i ), or a pure simplicial complexof dimension d − . In either case the decomposition in Theorem 3.2 finishes the proof. (cid:3) We can also see this via connections with the external activity complex defined by Ardilaand Boocher [1]. Following their convention, for subsets
S, T ⊆ [ n ] , we use x S y T to denotethe set { x i : i ∈ S } ∪ { y j : j ∈ T } . Definition 3.4. [1, Theorem 1.9] Let M be a matroid and suppose u ∈ R n has distinctcoordinates. Then u induces an order on [ n ] where i < j if and only if u i < u j . The externalactivity complex B u ( M ) is the simplicial complex on the ground set { x i , y i : i ∈ [ n ] } whoseminimal non-faces are { x min
With this translation of weights, we can realize the semi-broken circuit complex ∆ w ( M, I ) as the link of a face in the external activity complex B u ( M ) . Formally, the link of a face σ in the simplicial complex ∆ is the simplicial complex link ∆ ( σ ) = { τ ∈ ∆ : τ ∪ σ ∈ ∆ and τ ∩ σ = ∅} . It is the set of faces that are disjoint from σ but whose unions with σ lie in ∆ . Proposition 3.5.
Define weight vectors u, w ∈ R n as above. If the matroid M has no loopsin I , then the semi-broken circuit complex ∆ w ( M, I ) is isomorphic to the link of the face x I y [ n ] \ I in the external activity complex B u ( M ) .Proof. First we show that σ = x I y [ n ] \ I is actually a face of B u ( M ) by arguing that σ does notcontain the minimal non-face x min
The semi-broken circuit complex is shellable.Proof.
In [2], Ardila, Castillo, and Sampler show that the external activity complex, B u ( M ) ,is shellable. Then by [3, Prop. 10.14], the link of any face in B u ( M ) is also shellable. (cid:3) Example 3.7.
Let M be the matroid from Example 3.1, I = { , , } , and u be the weightvector associated to w as described above. It induces the linear order < < < < onthe ground set of the matroid M ( L ) .We outline the connection between the external activity complex B u ( M ) and the semi-broken circuit complex by tracking two bases B = { , , } , B = { , , } of the matroid M ( L ) in the construction of the two simplicial complexes. For each basis, we split the comple-ment [5] \ B i into externally active and externally passive elements. (See [1, §2.5] for the defini-tions of externally active and passive.) For B , { } is externally passive and { } is externallyactive. Then by [1, Theorem 5.1], the associated facet of B u ( M ) is F = x x x x y y y y . GEORGY SCHOLTEN AND CYNTHIA VINZANT
By deleting σ = x x x y y from F , we obtain the facet x y y of link ∆ ( σ ) , correspondingto the facet { , , } of ∆ w ( M, I ) . For B = { , , } , the externally passive elements are theentire complement { , } , hence the associated facet of B u ( M ) is F = x x x x x y y y .Since F does not contain σ , it does not contribute a facet to the link of σ in B u ( M ) .The connection between this simplicial complex and the semi-inverted linear space inv I ( L ) is that when w ∈ ( R + ) n has distinct coordinates, the ideal generated by the initial forms { in w ( f C ) : C is a circuit of M ( L ) } is the Stanley-Reisner ideal of ∆ w ( M, I ) . In fact, theinitial form of f C is in w ( f C ) = x b I ( C ) . The ideal generated by these initial forms is then theStanley-Reisner ideal I ∆ w ( M,I ) = (cid:104) in w ( f C ) : C ∈ C ( M ) (cid:105) .4. Proof of Theorem 1.1
In this section, we prove Theorem 1.1. To do this, we first use a flat degeneration of inv I ( L ) to establish a recursion for its degree. Proposition 4.1.
Suppose L is a linear subspace of C n and I ⊆ [ n ] . Let D ( L , I ) denote thedegree of the affine variety inv I ( L ) . (a) If i ∈ I is a loop of M ( L ) , then inv I ( L ) is empty and D ( L , I ) = 0 . (b) If i ∈ I is a co-loop of M ( L ) , then D ( L , I ) = D ( L /i, I \ i ) . (c) If i ∈ I is neither a loop nor a coloop of M ( L ) then D ( L\ i, I \ i ) + D ( L /i, I \ i ) ≤ D ( L , I ) . The proof of Theorem 1.1 will show that there is actually equality in part (c).
Proof.
Without loss of generality, take i = 1 .(a) If ∈ I is a loop of M ( L ) then L is contained in the hyperplane { x = 0 } . Thereforethe map inv I is undefined at every point of L and the image inv I ( L ) is empty. By convention,we take the degree of the empty variety to be zero.(b) If is a co-loop of M ( L ) , then L is a direct sum of span( e ) and L / , meaning thatany element in L can be written as ae + b where a ∈ C and b ∈ L / . For points at whichthe map inv I is defined, inv I ( ae + b ) = a − e + inv I \ ( b ) . From this, we see that inv I ( L ) isthe direct sum of span( e ) and inv I \ ( L / , implying that inv I ( L ) and inv I \ ( L / have thesame degree.(c) Let I denote the ideal of polynomials vanishing on inv I ( L ) and J = I denote itshomogenization in C [ x , x , . . . , x n ] . Take w = e ∈ R n +1 and consider in w ( J ) , as definedin Section 2.1. We will show that the variety of in w ( J ) contains the image in P n of both { } × inv I \ ( L\ and A ( C ) × inv I \ ( L / . Since both these varieties have dimension equalto dim( L ) , the degree of the variety of in w ( J ) is at least the sum of their degrees. The claimthen follows by the equality of the Hilbert series of J and in w ( J ) .If j ∈ I \ is a loop of M ( L ) , then j is a loop of M ( L\ and D ( L\ , I \
1) = 0 . Otherwisethe set U I is Zariski-dense in L , where U I denotes the intersection of L with ( C ∗ ) I × C [ n ] \ I .Let π I denote the coordinate projection C n → C I . On U I , the maps π I \ ◦ inv I and inv I \ ◦ π I \ are equal: π I \ (inv I ( x )) = inv I \ ( π I \ ( x )) = (cid:88) j ∈ I \ x − j e j + (cid:88) j (cid:54)∈ I x j e j . In particular, the points inv I \ ( π I \ ( U I )) are Zariski dense in inv I \ ( L\ . Now let x be apoint of inv I ( U I ) . Then [1 : x ] belongs to the variety of J and, for every t ∈ C , the point EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 9 ( t, t e · [1 : x ]) belongs to the variety of J w , as defined in Section 2.1. Taking t → , we seethat [1 : 0 : π I \ ( x )] belongs to the variety of in e ( J ) .If j ∈ I \ is a loop of M ( L / , then inv I \ ( L / is empty and the claim follows. Otherwisethe intersection U I \ of L / with { } × ( C ∗ ) I \ × C [ n ] \ I is nonempty and Zariski-dense in L ∩ { x ∈ C n : x = 0 } ∼ = L / . Let x ∈ U I \ . Since is not a loop of M ( L ) , there is a point v ∈ L with v = 1 . Then for any λ, t ∈ C ∗ , x + ( t/λ ) v belongs to L and for all but finitelymany values of t , y ( t ) = inv I ( x + ( t/λ ) v ) is defined and has first coordinate y ( t ) = λ/t .Then [1 : y ( t )] ∈ V ( J ) and ( t, t e · [1 : y ( t )]) belongs to V (cid:0) J w (cid:1) . Note that the limit of t e · [1 : y ( t )] = [1 : λ : y ( t ) : . . . : y n ( t )] as t → equals [1 : λ : inv I \ ( x )] . Therefore forevery point ( λ, u ) ∈ A ( C ) × inv I \ ( L / , the point [1 : λ : u ] belongs to V (in e ( J )) . (cid:3) We also need the following fact from commutative algebra, included here for completeness,which may be clear to readers familiar with the geometry of schemes. Recall that for ahomogeneous ideal J ⊂ C [ x ] , the Hilbert polynomial of J is the polynomial h ( t ) that agreeswith dim C ( C [ x ] /J ) t for sufficiently large t ∈ N . Then h ( t ) = (cid:80) di =0 b i (cid:0) td − i (cid:1) for some d ∈ Z > ,where b i ∈ Z with b > . In a slight abuse of notation, we say that the dimension of J is dim( J ) = d and the degree of J is deg( J ) = b . The ideal J is equidimensional of dimension d if dim( P ) = d for every minimal associated prime P of J . Lemma 4.2.
Let I ⊆ J ⊆ C [ x ] be equidimensional homogeneous ideals of dimension d . If I is radical and deg( I ) ≤ deg( J ) , then I and J are equal.Proof. Let I = P ∩ . . . ∩ P r and J = Q ∩ . . . ∩ Q s be irredundant primary decompositionsof I and J . Without loss of generality, we can assume that dim( Q i ) = d for ≤ i ≤ u ,and since V ( J ) ⊆ V ( I ) , the prime ideals P i can be reindexed such that P i = √ Q i , implying Q i ⊆ P i . For all ≤ i ≤ u , there exists an element a ∈ ( ∩ j (cid:54) = i P j ) ∩ ( ∩ j (cid:54) = i (cid:112) Q j ) with a (cid:54)∈ P i .Then the saturation I : (cid:104) a (cid:105) ∞ = P i is contained in J : (cid:104) a (cid:105) ∞ = Q i , implying P i = Q i . Thiswrites the ideal J as J = P ∩ . . . ∩ P u ∩ Q u +1 ∩ . . . ∩ Q s . The degree of an ideal is equal tothe sum of the degrees of the top dimensional ideals in its primary decomposition, hence deg( I ) = r (cid:88) i =1 deg( P i ) and deg( J ) = u (cid:88) i =1 deg( Q i ) = u (cid:88) i =1 deg( P i ) . The assumption that deg( I ) ≤ deg( J ) implies that r = u , which gives the reverse contain-ment I = P ∩ . . . ∩ P u ⊇ J . (cid:3) Proof of Theorem 1.1.
We proceed by induction on | I | . If | I | = 0 , then inv I ( L ) is just thelinear space L . Then Theorem 1.1 reduces to the statement that the linear forms supportedon circuits form a universal Gröbner basis for I ( L ) . See e.g. [17, Prop. 1.6].Now take | I | ≥ , w ∈ ( R + ) n with distinct coordinates, and let M denote the matroid M ( L ) . If M has a loop i in I , then for the circuit C = { i } , the circuit polynomial f C equals , which is a Gröbner basis for the ideal of polynomials vanishing on the empty set inv I ( L ) .Therefore we may suppose that M has no loops in I , in which case inv I ( L ) is a d -dimensionalaffine variety of degree D ( L , I ) .Let ∆ denote the I -broken circuit complex ∆ w ( M, I ) defined in Section 3 and let ∆ denote the simplicial complex on elements { , . . . , n } obtained from ∆ by coning over thevertex . Let I ∆ denote the Stanley-Reisner ideal of ∆ , as in Section 2.2.Let I ⊂ C [ x ] be the ideal of polynomials vanishing on inv I ( L ) and define the ideal J ⊂ C [ x , x , . . . , x n ] to be its homogenization with respect to x . Since inv I ( L ) is the image of an irreducible variety under a rational map, it is also irreducible. It follows that the ideals I and J are prime. For a circuit polynomial f C , its homogenization f C belongs to J andsince w ∈ ( R + ) n , in (0 ,w ) ( f C ) = in w ( f C ) = (cid:40) a k x C \ k if C ⊆ I and k = argmin { w j : j ∈ C } a k x C ∩ I ∪ k if C (cid:54)⊆ I and k = argmax { w j : j ∈ C \ I } .Up to a scalar multiple, in w ( f C ) equals the square-free monomial corresponding to the I -broken circuit of C , namely x b I ( C ) . It follows that (cid:104) in w ( f C ) : C ∈ C ( M ) (cid:105) = I ∆ and (cid:104) in (0 ,w ) ( f C ) : C ∈ C ( M ) (cid:105) = I ∆ . From this we see that I ∆ ⊆ in (0 ,w ) ( J ) .Let i = argmax { w j : j ∈ I } . By the inductive hypothesis, D ( L\ i, I \ i ) and D ( L /i, I \ i ) are the number of facets of ∆ w ( M \ i, I \ i ) and ∆ w ( M/i, I \ i ) , respectively. Therefore byTheorem 3.2, ∆ and thus ∆ each have D ( L /i, I \ i ) facets if i is a coloop of M and D ( L\ i, I \ i ) + D ( L /i, I \ i ) facets otherwise. Then by Proposition 4.1, ∆ has at most D ( L , I ) facets and the Stanley-Reisner ideal I ∆ has degree ≤ D ( L , I ) .Since ∆ is a pure simplicial complex of dimension d , I ∆ is an equidimensional ideal of di-mension d . As J is a prime d -dimensional ideal, its initial ideal in (0 ,w ) ( J ) is equidimensionalof the same dimension, see [8, Lemma 2.4.12].The ideals I ∆ and in (0 ,w ) ( J ) then satisfy the hypotheses of Lemma 4.2, and we concludethat they are equal. By [8, Prop. 2.6.1], restricting to x = 1 gives that I ∆ = (cid:104) in w ( f C ) : C ∈ C ( M ) (cid:105) = in w ( I ) . As this holds for every w ∈ ( R + ) n with distinct coordinates, it will also hold for arbitrary w ∈ ( R ≥ ) n (see [17, Prop. 1.13]). It follows from [17, Cor. 1.9, 1.10] that the circuitpolynomials { f C : C ∈ C ( M ) } form a universal Gröbner basis for I . (cid:3) Example 4.3.
Consider the -dimensional linear space in C : L = rowspan . The circuits of the matroid M ( L ) are C = { , , } . Take I = { , , } . Then f = x + x − x x x , f = x + x − x x x , and f = x − x + x x x − x x x . If w ∈ ( R + ) with w < . . . < w , then the ideal (cid:104) in w ( f C ) : C ∈ C(cid:105) is (cid:104) x x x , x x x , x x x (cid:105) .The simplicial complex ∆ w ( M, I ) is -dimensional and has seven facets: facets(∆ w ( M, I )) = { , , , , , , } . Indeed, the variety of (cid:104) x x x , x x x , x x x (cid:105) is the union the seven coordinate linear spaces span { e i , e j , e k } where { i, j, k } is a facet of ∆ w ( M, I ) .Interestingly, it is not true that the homogenizations f C form a universal Gröbner basis forthe homogenization I . Indeed, consider the weight vector (2 , , , , , . The ideal generatedby the initial forms of circuit polynomials (cid:104) in w ( f C ) : C ∈ C(cid:105) is (cid:104) x x + x x , x x (cid:105) , whereas in w ( I ) = (cid:104) x x x − x x x − x x x , x x + x x , x x (cid:105) . Nevertheless, upon restriction to x = 1 , the two ideals become equal. EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 11
Corollary 4.4. If dim( L ) = d , then the affine Hilbert series of the ideal
I ⊆ C [ x ] ofpolynomials vanishing on inv I ( L ) is ∞ (cid:88) m =0 dim C ( C [ x ] ≤ m / I ≤ m ) t m = 1(1 − t ) d +1 d (cid:88) i =0 f i − t i (1 − t ) d − i = h + h t + . . . + h d t d (1 − t ) d +1 . where ( f − , . . . , f d − ) and ( h , . . . , h d ) are the f - and h -vectors of ∆ w ( M, I ) . In particular,its degree is the number of facets f d − = h + h + . . . + h d .Proof. The affine Hilbert series of I equals the classical Hilbert series of its homogenization I , which equals the Hilbert series of in (0 ,w ) ( I ) for any w ∈ R n . When the coordinates of w are distinct and positive, in (0 ,w ) ( I ) is the Stanley-Reisner ideal of ∆ = cone(∆ w ( M, I ) , .Since the Stanley-Reisner ideals of ∆ = ∆ w ( M, I ) and ∆ are generated by the same square-free monomials, their Hilbert series differ by a factor of / (1 − t ) . The result then followsfrom well known formulas for the Hilbert series of I ∆ , [10, Ch. 1]. (cid:3) The proof of Theorem 1.1 shows that there is equality in Proposition 4.1(c), namelythat if i ∈ I is neither a loop nor a coloop of M ( L ) , then the degree D ( L , I ) satisfies D ( L , I ) = D ( L\ i, I \ i ) + D ( L /i, I \ i ) . From this we can derive an explicit formula for thedegree of inv I ( L ) in the uniform matroid case. Corollary 4.5.
For a generic d -dimensional linear space L ⊆ C n and I ⊆ [ n ] of size | I | = k ,the degree of inv I ( L ) equals D ( L , I ) = d (cid:88) j = k + d − n (cid:18) kj (cid:19) − (cid:18) k − d (cid:19) , where we take (cid:0) ab (cid:1) = 0 whenever a < or b < . In particular, for n ≥ k + d , the degree onlydepends on d and k .Proof. By assumption k, d, n satisfy the inequalities ≤ k ≤ n and ≤ d ≤ n . We proceedby induction on k . In the extremal cases, D ( L , I ) satisfies D ( L , I ) = if k = 0 , if d = n, if d = 0 and k ≥ . Indeed, if I = ∅ , then inv I ( L ) = L and D ( L , I ) = 1 . If d = n , then inv I ( L ) is all of C n and D ( L , I ) = 1 . Finally, if d = 0 and | I | ≥ , then n ≥ | I | ≥ , and L = { (0 , . . . , } in C n .The map inv I is not defined at this point so inv I ( L ) is empty and thus has degree .Suppose k ≥ and < d < n . Since any i ∈ I is neither a loop nor a coloop, D ( L , I ) equals D ( L\ i, I \ i ) + D ( L /i, I \ i ) by Proposition 4.1(c) and the proof of Theorem 1.1. Recallthat L\ i and L /i are subspaces in C n − of dimensions d and d − , respectively. Since | I \ i | = k − , by induction we get that D ( L\ i, I \ i ) = d (cid:88) j = k + d − n (cid:18) k − j (cid:19) − (cid:18) k − d (cid:19) , and D ( L /i, I \ i ) = d − (cid:88) j = k + d − n − (cid:18) k − j (cid:19) − (cid:18) k − d − (cid:19) . Since (cid:0) k − j (cid:1) + (cid:0) k − j − (cid:1) = (cid:0) kj (cid:1) and (cid:0) k − d (cid:1) + (cid:0) k − d − (cid:1) = (cid:0) k − d (cid:1) , the sum D ( L\ i, I \ i ) + D ( L /i, I \ i ) isthe desired formula for D ( L , I ) . (cid:3) Example 4.6.
The number of facets of the complex ∆ w ( M, I ) gives the degree D ( L , I ) andif M is the uniform matroid of rank d on [ n ] , we can write out these facets explicitly. Let w = (1 , . . . , n ) and consider I = { , . . . , k } . If k ≤ d , no circuit is contained in the invertedset I , implying that every broken circuit has the form ( C ∩ I ) ∪ max { C \ I } . Then everyfacet of ∆ w ( M, I ) has the form S ∪ { k + 1 , . . . , k + d − j } where S ⊆ I and | S | = j ≤ d .For fixed j , the number of possibilities are (cid:0) kj (cid:1) , and the constraints on j are k + d − j ≤ n and ≤ j ≤ k ≤ d . If k > d , then every subset of { , . . . , k } of size d is an I -broken circuit.From the list of facets S ∪ { k + 1 , . . . , k + d − j } , we remove those for which S ⊂ { , . . . , k } and | S | = d , of which there are (cid:0) k − d (cid:1) . 5. Supports
In this section, we characterize the intersection of the variety inv I ( L ) with the coordinatehyperplanes. These are exactly the points in the closure of, but not the actual image of, themap inv I . Given a point p ∈ C n , its support is the set of indices of its nonzero coordinates: supp( p ) = { i : p i (cid:54) = 0 } . For a subset S ⊆ [ n ] , we will use C S to denote the set of points p with supp( p ) ⊆ S and S to denote the complement [ n ] \ S . Theorem 5.1.
Suppose that the matroid M = M ( L ) has no loops in I . For S ⊆ [ n ] , let T = S ∪ I . If T is a flat of M , then the restriction of inv I ( L ) to C S is given by inv I ( L ) ∩ C S = inv S ∩ I (cid:0) π T ( L ) ∩ C S (cid:1) , where π T denotes the coordinate projection C n → C T . Moreover, supp( p ) = S for some p ∈ inv I ( L ) if and only if T is a flat of M and T \ S is a flat of M | T . We build up to the proof of Theorem 5.1 by considering the cases I ⊆ S and I ⊆ S . Lemma 5.2. If S ⊆ [ n ] is a flat of M with I ⊆ S , then inv I ( L ) ∩ C S = inv S ∩ I ( π S ( L )) , where π S denotes the coordinate projection C n → C S .Proof. Recall that F ⊂ [ n ] is a flat of M if and only if | F ∩ C | (cid:54) = 1 for all circuits C of M .Suppose that S is a flat of M and consider the restriction of the circuit polynomials f C to C S . Note that S ⊆ I , so that for any circuit C with | C ∩ S | ≥ , | C ∩ I | ≥ and the circuitpolynomial f C is zero at every point of C S .The circuits for which | C ∩ S | = 0 are exactly the circuits contained in S , which arethe circuits of the matroid restriction M | S . Moreover the projection π S ( L ) is cut out bythe vanishing of the linear forms { (cid:96) C : C ∈ C ( M ) , C ⊆ S } , which are exactly the linearforms { (cid:96) C (cid:48) : C (cid:48) ∈ C ( M | S ) } . It follows that the circuit polynomials { f C (cid:48) : C (cid:48) ∈ C ( M | S ) } area subset of the circuit polynomials of L , namely { f C : C ∈ C ( M ) , C ⊆ S } . By Theorem 1.1,the variety of circuit polynomials is the variety of the semi-inverted linear space, giving that inv I ( L ) ∩ C S = V ( { f C : C ∈ C ( M ) , C ⊆ S } ) ∩ C S = V ( { f C (cid:48) : C (cid:48) ∈ C ( M | S ) } ) = inv S ∩ I ( π S ( L )) . (cid:3) EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 13
Lemma 5.3. If S ⊆ [ n ] with I ⊆ S , then inv I ( L ) ∩ C S = inv I (cid:0) L ∩ C S (cid:1) .Proof. ( ⊇ ) The affine variety inv I (cid:0) L ∩ C S (cid:1) is the Zariski-closure of L ∩ C S under the map inv I . Since L ∩ C S is contained in L , inv I (cid:0) L ∩ C S (cid:1) is a subset of inv I ( L ) . Moreover inv I ( p ) ∈ C S for any point p ∈ C S . The inclusion follows.( ⊆ ) For this we show the reverse inclusion of the ideals of polynomials vanishing on thesevarieties. Now let C (cid:48) be a circuit of M ( L ∩ C S ) and (cid:96) C (cid:48) = (cid:80) i ∈ C (cid:48) a i x i its corresponding linearform. Then for some circuit C of M , C (cid:48) = C ∩ S and (cid:96) C (cid:48) equals the restriction (cid:96) C ( π S ( x )) .Applying inv I and clearing denominators then gives f C (cid:48) ( x ) = x C (cid:48) ∩ I (cid:96) C (cid:48) (inv I ( x )) = x C ∩ I (cid:96) C (inv I ( π S ( x ))) = f C ( π S ( x )) . The middle equation holds because I ⊆ S , which implies that C \ C (cid:48) ⊆ S ⊆ I . (cid:3) Proof of Theorem 5.1.
Suppose that T is a flat of M . Since I ⊆ T , Lemma 5.2 says that therestriction inv I ( L ) | C T equals inv T ∩ I ( π T ( L )) . Furthermore since T ∩ I = S ∩ I ⊆ S , we canapply Lemma 5.3 to find the intersection of inv T ∩ I ( π T ( L )) with C S . All together this givesthat inv I ( L ) ∩ C S equals(3) (inv I ( L ) ∩ C T ) ∩ C S = inv S ∩ I ( π T ( L )) ∩ C S = inv S ∩ I (cid:0) π T ( L ) ∩ C S (cid:1) . Suppose further that T \ S is a flat of the matroid M | T . This implies that the contractionof M | T by T \ S has no loops. This is the matroid of the linear space π T ( L ) ∩ C S , whichis therefore not contained in any coordinate subspace { x i = 0 } for i ∈ S . It follows thatthere is a point p ∈ π T ( L ) ∩ C S of full support supp( p ) = S . Equation (3) then shows that inv S ∩ I ( p ) is a point of support S in inv I ( L ) .Conversely, suppose that S = supp( p ) for some point p ∈ inv I ( L ) . Then T is a flat of M .To see this, suppose for the sake of contradiction that for some circuit C of M , C ∩ T = { j } .Then j is the unique element of C ∩ I for which p j = 0 , and evaluating the circuit polynomial f C at the point p gives f C ( p ) = (cid:88) i ∈ C ∩ I a i p C ∩ I \{ i } + (cid:88) i ∈ C \ I a i p C ∩ I ∪{ i } = a j p C ∩ I \{ j } (cid:54) = 0 , contradicting p ∈ inv I ( L ) . Therefore T is a flat of M and (3) holds. It follows that p , ormore precisely π T ( p ) , is a point of support S in π T ( L ) . Therefore π T ( L ) ∩ C S contains apoint of full support, the contraction of the matroid M | T by T \ S has no loops, and T \ S isa flat of the matroid M | T . (cid:3) Example 5.4.
Suppose L is a generic d -dimensional subspace of C n , and hence that M = M ( L ) is the uniform matroid of rank d on [ n ] . Its flats are the subsets F ⊆ [ n ] of size | F | < d , along with the full set [ n ] . Consider S ⊆ [ n ] and T = S ∪ I . If T is a flat of M , theneither | T | < d , implying | I | < d , or T = [ n ] , in which case I ⊆ S . If | T | < d , then M | T isthe uniform matroid of rank | T | on the elements T . Then every subset of T is a flat of M | T and S is the support of a point in inv I ( L ) . If T = [ n ] , then T \ S = S is a flat of M | T = M ifand only if | S | < d or | S | = n . Since S contains I , | S | = n only when I = S = ∅ . Thereforeif I (cid:54) = ∅ , we have | S | > n − d . Putting these together gives S ∈ supp(inv I ( L )) ⇐⇒ S = ∅ or | S | > n − d if I = ∅ I ⊆ S and | S | > n − d if < | I | ≤ n − d | S ∪ I | < d or I ⊆ S if n − d < | I | . Real points and hyperplane arrangements
Here we explore a slight variation of inv I that preserves a real-rootedness property ofcertain intersections. Given a polynomial f ∈ R [ x , . . . , x n ] with a real-rootedness propertycalled stability , it is known that the polynomial x deg e ( f )1 · f ( − /x , x , . . . , x n ) is again stable[20, Lemma 2.4]. Here we extend that to an action preserving real-rootedness of intersectionswith a family of affine-spaces. For I ⊆ [ n ] , define the rational map inv − I : C n (cid:57)(cid:57)(cid:75) C n by (inv − I ( x )) i = (cid:40) − /x i if i ∈ Ix i if i (cid:54)∈ I. Equivalently this is the composition of inv I with the map that scales coordinates x i for i ∈ I by − . Note that the varieties inv I ( L ) and inv − I ( L ) are isomorphic, and in particular theyhave the same degree. For any linear space L ⊂ C n , let L ⊥ denote the subspace of vectors v for which (cid:80) ni =1 v i x i = 0 for all x ∈ L . Proposition 6.1. If L ⊂ C n is invariant under complex conjugation, then for any u ∈ R n ,all of the intersection points of inv − I ( L ) with L ⊥ + u are real.Proof. If L is contained in a coordinate hyperplane { x i = 0 } where i ∈ I , then inv − I ( L ) isempty and the claim trivially follows. Otherwise, the points x ∈ inv − I ( L ) with x i (cid:54) = 0 for i ∈ I are necessarily Zariski-dense, and for a generic point u ∈ R n , the intersection points of inv − I ( L ) with L ⊥ + u belongs to ( C ∗ ) I × C [ n ] \ I . Showing that these intersection points arereal for generic u implies it for all.Suppose that a point a + i b belongs to the intersection of inv − I ( L ) with L ⊥ + u where a, b ∈ R n and a j + i b j (cid:54) = 0 for every j ∈ I . Then ( a − u ) + i b belongs to L ⊥ . Since L ⊥ isconjugation invariant, it follows that b ∈ L ⊥ . In particular, b T x = 0 for all x ∈ L . Since a + i b belongs to inv − I ( L ) , inv − I ( a + i b ) belongs to L . It follows that b T inv − I ( a + i b ) = 0 .Taking imaginary parts gives (cid:32)(cid:88) j ∈ I − b j a j + i b j + (cid:88) j (cid:54)∈ I b j ( a j + i b j ) (cid:33) = (cid:88) j ∈ I b j a j + b j + (cid:88) j (cid:54)∈ I b j . Since every term is nonnegative and their sum is zero, each term must be zero. Thus b j = 0 for all j and the point a + i b is real. (cid:3) Remark 6.2.
Propostion 6.1 shows that inv − I ( L ) is hyperbolic with respect to L ⊥ , in thesense of [15]. In fact, one can replace L ⊥ in this statement by any linear space of the samedimension whose non-zero Plücker coordinates agree in sign with those of L ⊥ . This showsthat inv − I ( L ) is a stable variety . See [7, Section 2] for more. Proposition 6.3.
For generic u ∈ R n , the intersection points of inv − I ( L ) with L ⊥ + u arethe minima of the function (4) f ( x ) = 12 (cid:88) j (cid:54)∈ I x j − (cid:88) j ∈ I log | x j | over the regions in the complement of the ( affine ) hyperplane arrangement { x i = 0 } i ∈ I inthe affine linear space L ⊥ + u .Proof. On ( R ∗ ) I × R [ n ] \ I , f is infinitely differentiable and we examine its behavior on eachorthant. For a sign pattern σ : I → {± } , let R Iσ denote the orthant of points in ( R ∗ ) I with EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 15 σ ( i ) x i > for all i ∈ I . Inspecting the Hessian of f shows that it is also strictly convex on R Iσ × R [ n ] \ I . Indeed, the Hessian of f is a diagonal matrix whose ( j, j ) th entry is equal to /x j for j ∈ I and for j (cid:54)∈ I and is therefore positive definite on ( R ∗ ) I × R [ n ] \ I .Define the (open) polyhedron P σ to be the intersection of R Iσ × R [ n ] \ I with the affine space L ⊥ + u . The function f is strictly convex on P σ . Therefore any critical point of f over P σ isa global minimum. The affine span of P σ is L ⊥ + u , so p ∈ P σ is a critical point of f when ∇ f ( p ) belongs to ( L ⊥ ) ⊥ = L . Since ∇ f ( p ) = inv − I ( p ) and inv − I is an involution, this impliesthat p belongs to inv − I ( L ) . Putting this all together, we find that for a point p ∈ P σ , p attains the minimum of f over P σ ⇔ ∇ f ( p ) ∈ L ⇔ p ∈ inv − I ( L ) . (cid:3) We can characterize which connected components of ( L ⊥ + u ) \{ x i = 0 } i ∈ I contains a pointin inv − I ( L ) in terms of the recession cone rec( P σ ) = ( R Iσ × R [ n ] \ I ) ∩ L ⊥ . Lemma 6.4.
The infimum of f over P σ is attained if and only if the intersection of R I withthe recession cone of P σ is trivial, i.e. rec( P σ ) ∩ R I = { } .Proof. ( ⇒ ) Suppose rec( P σ ) ∩ R I contains v (cid:54) = 0 . Then for any p ∈ P σ , the univariatefunction f ( p + tv ) = (cid:80) j (cid:54)∈ I p j − (cid:80) i ∈ I log | p i + tv i | is strictly decreasing as t → ∞ and theinfimum of f is not attained on P σ .( ⇐ ) Suppose that rec( P σ ) ∩ R I = { } . Then the quadratic form (cid:80) j (cid:54)∈ I x j is positive definiteon the recession cone rec( P σ ) . We can write P σ as Q + rec( P σ ) , where Q is a compactpolytope. Let S denote the section of the recession cone, S = { v ∈ rec( P σ ) : || v || = 1 } . Forany point p ∈ Q and v ∈ S , consider the univariate function t (cid:55)→ f ( p + tv ) , which is strictlyconvex and continuous on { t : p + tv ∈ P σ } . Its derivative ddt f ( p + tv ) = (cid:88) j (cid:54)∈ I v j p j + t (cid:88) j (cid:54)∈ I v j − (cid:88) i ∈ I v i p i + tv i has a unique root t ∈ R for p + tv ∈ P σ . Indeed, by assumption (cid:80) j (cid:54)∈ I v j > . Then, since d dt f ( p + tv ) > where defined, ddt f ( p + tv ) is strictly increasing on { t : p + tv ∈ P σ } . If v ∈ R [ n ] \ I , then this set is all of R and ddt f ( p + tv ) is linear. Otherwise, there is a minimum t for which p + tv ∈ P σ and ddt f ( p + tv ) → −∞ as t approaches this minimum, whereas ddt f ( p + tv ) > for sufficiently large t . Let t ∗ ( p, v ) denote this unique root of ddt f ( p + tv ) . Thisis a continuous function in p and v . Let T denote the maximum of t ∗ ( p, v ) over ( p, v ) ∈ Q × S .Now we claim that when minimizing f over P σ , it suffices to minimize over the compact set Q + [0 , T ] S . Indeed, if y ∈ P σ , then y = p + tv for some p ∈ Q , v ∈ S and t ∈ R > . If t > T ,then the point x = p + T v ∈ Q + [0 , T ] S satisfies f ( x ) < f ( y ) . In particular, the minimumof f is bounded from below and is therefore attained on the compact set Q + [0 , T ] S . (cid:3) Proposition 6.5.
For generic u ∈ R n , there is exactly one point of inv − I ( L ) in each regionof ( L ⊥ + u ) \{ x i = 0 } i ∈ I whose recession cone has trivial intersection with R I . The degree of inv − I ( L ) equals the number of these regions.Proof. First we show that for generic u ∈ R n , the number of intersection points of inv − I ( L ) with L ⊥ + u equals the degree of inv − I ( L ) . To do this, we show that the closure inv − I ( L ) in P n ( C ) has no points in common with L ⊥ + x u with x = 0 . For the sake of contradictionsuppose that for some a ∈ L ⊥ , the point [0 : a ] belongs to inv − I ( L ) and let S = supp( a ) . It follows that a T x = (cid:80) i ∈ S a i x i vanishes on L , g = x S ∩ I · a T inv − I ( x ) vanishes on inv − I ( L ) ,and the homogenezation g hom with respect to x vanishes on the closure inv − I ( L ) ⊆ P n ( C ) .In particular, g hom (0 , a ) = 0 . If S ⊆ I , this contradicts the evaluation of the polynomial g hom = g = (cid:80) j ∈ S a j x S \ j given by g hom (0 , a ) = (cid:88) j ∈ S a S = a S · | S | (cid:54) = 0 . Similarly, since inv − I ( L ) is invariant under complex conjugation, we also have g hom (0 , a ) = 0 ,where a is the complex conjugate of a . If S (cid:54)⊆ I , this contradicts the evaluation of thepolynomial g hom = − x (cid:80) j ∈ S ∩ I a j x S ∩ I \ j + x S ∩ I (cid:80) j ∈ S \ I a j x j given by g hom (0 , a ) = a S ∩ I (cid:88) j ∈ S \ I a j a j (cid:54) = 0 . Therefore all the intersection points of inv − I ( L ) with L ⊥ + x u have x (cid:54) = 0 . Then for generic u , the number of intersection points of inv − I ( L ) and L ⊥ + u equals the degree of inv − I ( L ) .By Propositions 6.1 and 6.3, each of these intersection points is real and thus is a minimizerof the function f ( x ) of (4) over some connected component P σ of ( L ⊥ + u ) \{ x i = 0 } i ∈ I . ByLemma 6.4, the components P σ contains a minimizer if and only if rec( P σ ) ∩ R I = { } . (cid:3) This together with Corollary 4.4 constitutes the proof of Theorem 1.2. For special casesof I , we find a simpler characterization of the regions counted by deg(inv I ( L )) . Corollary 6.6.
Let u ∈ R n be generic. If I is independent in the matroid M ( L ) , then thedegree of inv I ( L ) equals the total number of regions in ( L ⊥ + u ) \{ x i = 0 } i ∈ I . If I = [ n ] , thenthe degree of inv I ( L ) equals the number of bounded regions in ( L ⊥ + u ) \{ x i = 0 } i ∈ I .Proof. If I is independent in M ( L ) , then I is contained in a basis B of M ( L ) , and [ n ] \ B isa basis of M ( L ⊥ ) contained in [ n ] \ I . In particular, if x ∈ L ⊥ has x j = 0 for all j ∈ [ n ] \ I ,then x = 0 . So R I ∩ L ⊥ = { } . The recession cone of any region of ( L ⊥ + u ) \{ x i = 0 } i ∈ I iscontained in L ⊥ , so its intersection with R I is trivial.If I = [ n ] , then R I = R n . The recession cone of a region in ( L ⊥ + u ) \{ x i = 0 } i ∈ I containsa non-zero vector if and only if it is unbounded. Therefore the regions whose recession coneshave trivial intersection with R I are those which are bounded. (cid:3) Example 6.7.
Consider the -dimensional linear space L from Example 4.3 and take thevector u = (0 , , , , . The two-dimensional affine space L ⊥ + u consists of points of theform ( x , x , x − x + 1 , − x + 2 , − x + x + 2) . Since I = { , , } is independent in M ( L ) ,each of the seven regions in the complement of the hyperplane arrangement { x i = 0 } i ∈ I contains a point of inv − I ( L ) . For I = { , , , } , there are four regions whose recession conesintersect { x = 0 } nontrivially. The remaining six regions each contain a unique point in inv − I ( L ) . Finally, for I = { , , , , } , R I is all of R so the recession cone of P σ intersects R I nontrivially if and only if P σ is unbounded. Thus the four bounded regions of the hyperplanearrangement { x i = 0 } i ∈ I in L ⊥ + u are precisely those that contain points in inv − I ( L ) . Thesehyperplane arrangements and intersection points are shown in Figure 1. EMI-INVERTED LINEAR SPACES AND AN ANALOGUE OF THE BROKEN CIRCUIT COMPLEX 17
Figure 1.
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