Lorentzian polynomials from polytope projections

Lorentzian polynomials, recently introduced by Br\"and\'en and Huh, generalize the notion of log-concavity of sequences to homogeneous polynomials whose supports are integer points of generalized permutahedra. Br\"and\'en and Huh show that normalizations of polynomials equaling integer point transforms of generalized permutahedra are Lorentzian; moreover, normalizations of certain projections of integer point transforms of generalized permutahedra with zero-one vertices are also Lorentzian. Taking this polytopal perspective further, we show that normalizations of certain projections of integer point transforms of flow polytopes (which, before projection, are not Lorentzian), are also Lorentzian.


Introduction
Log-concavity of sequences is a classical notion, which often is either very easy or notoriously difficult to prove. A sequence a 0 , a 1 , . . . , a n is said to be log-concave if a 2 i ≥ a i−1 a i+1 for i ∈ [n − 1]. In groundbreaking recent work Brändén and Huh [BH19] introduced Lorentzian polynomials (see Section 2.1 for definition), which generalize the notion of log-concavity. Just one of their theory's many consequences are the celebrated Alexandrov-Fenchel inequalities on mixed volumes of Minkowski sums of polytopes; these inequalities follow from the volume polynomial being Lorentzian [BH19, Theorem 9.1].
Our motivation for the present paper is simple: (whenever possible) understand Lorentzian polynomials polytopally. Brändén and Huh show that the support of a Lorentzian polynomial form the integer points of a generalized permutahedron, a beautiful polytope studied extensively by Postnikov in [Pos09].
Recall that for a polytope P ⊂ R n , the integer point transform of P is defined as (1.1) σ P (x 1 , . . . , x n ) = p∈P ∩Z n x pi i .
Define the normalization operator N on R[x 1 , . . . , x n ] by where for a vector α = (α 1 , . . . , α n ) of nonnegative integers we write α! to mean n i=1 α i !. Brändén  The current paper adds a natural class of polytope/projection pairs yielding Lorentzian polynomials: flow polytopes/projection onto a coordinate hypersurface.
The flow polytope F G (a) associated to a loopless graph G on vertex set [n + 1] with edges directed from smaller to larger vertices and to the netflow vector a = (a 1 , . . . , a n+1 ) ∈ Z n+1 is: ≥0 : M G f = a}, where M G is the incidence matrix of G; that is, the columns of M G are the vectors e i − e j for (i, j) ∈ E(G), i < j, where e i is the i-th standard basis vector in R n+1 . The points f ∈ F G (a) are called (a-)flows (on G).
Observe that the number of integer points in F G (a) is the number of ways to write a as a nonnegative integral combination of the vectors e i − e j for edges (i, j) in G, i < j, which we refer to as the Kostant partition function K G (a).
We define two natural projections ϕ and ψ of F G (a) onto generalized permutahedra in Propositions 3.4 and 3.6 in Section 3. The projections ϕ and ψ induce projections on the integer point transform σ F G (a) (x) of F G (a), acting on monomials via x f → x ϕ(f ) and x f → x ψ(f ) . The resulting projected polynomials are denoted While the normalization of the integer point transform of F G (a) is not Lorentzian in general, we prove that the normalizations of its projections σ ϕ G(a) and σ ψ G(a) are always Lorentzian: Theorem 5.1. Let G be a loopless directed graph on the vertex set [n + 1] with a unique sink, and let a = (a 1 , . . . , a n+1 ) ∈ Z n ≥0 × Z ≤0 . The polynomials N (σ ϕ G(a) ) and N (σ ψ G(a) ) are Lorentzian. Theorem 5.1 implies that the Kostant partition function is log-concave along root directions (Corollary 5.2). We remark that log-concavity of the Kostant partition function along root directions is also a corollary of volume polynomials (of flow polytopes) being Lorentzian (Theorem 2.7).
Roadmap of the paper. Section 2 contains the necessary background on Lorentzian polynomials, generalized permutahedra and flow polytopes. Section 3 introduces the projections ϕ and ψ of F G (a) onto generalized permutahedra that we are interested in, while Section 4 studies their fibers. Section 5 establishes our main result, Theorem 5.1. Section 6 prods Question 1.1.

Background
In this section we give background on the main players of the paper: Lorentzian polynomials, generalized permutahedra and flow polytopes.
2.1. Lorentzian polynomials and generalized permutahedra. Let N = {0, 1, 2, . . .}, and denote by e i the ith standard basis vector of N n . A subset J ⊆ N n is called M-convex if for any index i and any α, β ∈ J whose ith coordinates satisfy α i > β i , there is an index j satisfying α j < β j , α − e i + e j ∈ J, and β − e j + e i ∈ J.
The convex hull of an M-convex set is a polytope also called a generalized permutahedron. A special class of generalized permutahedra consist of Minkowski sums of scaled coordinate simplices: for a subset S ⊆ [n], the coordinate simplex ∆ S ⊆ R n is the convex hull of the coordinate basis vectors {e i } i∈S . Minkowski sums of scaled coordinate simplices are called y-generalized permutahedra.
Let H d n be the space of degree d homogeneous polynomials with real coefficients in the n variables x 1 , . . . , x n . For f ∈ H d n , we write supp(f ) ⊆ N n for the support of f . For f ∈ H d n , denote by ∂ ∂xi f the partial derivative of f relative to x i . The Hessian of a homogenous quadratic polynomial f ∈ H 2 n is the symmetric n × n matrix H = (H ij ) i,j∈[n] defined by H ij = ∂ i ∂ j f . The set L d n of Lorentzian polynomials with degree d in n variables is defined as follows. Set L 1 n ⊆ H 1 n to be the set of all linear polynomials with nonnegative coefficients. Let L 2 n ⊆ H 2 n be the subset of quadratic polynomials with nonnegative coefficients whose Hessians have at most one positive eigenvalue and which have M-convex support. For d > 2, define L d n ⊆ H d n recursively by L d n = f ∈ M d n : where M d n ⊆ H d n is the set of polynomials with nonnegative coefficients whose supports are M-convex.
Recall the normalization operator N on R[x 1 , . . . , x n ]: where for a vector α = (α 1 , . . . , α n ) of nonnegative integers we write α! to mean observe that the ij-th entry of the Hessian of N (f ), namely the quantity We arrive at the following criterion for Lorentzian polynomials. It is our main workhorse in the proof of Theorem 5.1. For each d = (d 1 , . . . , d n ) with d 1 + · · · + d n = d − 2 and d i ∈ Z ≥0 for i ∈ [n], define the n × n matrix consisting of coefficients of f . Then N (f ) ∈ L d n if and only if H d has at most one positive eigenvalue for each d.
Proof. Note that normalization and differentiation preserve M -convexity of the support of a polynomial. By Lemma 2.2, we obtain is Lorentzian if and only if H d has at most one positive eigenvalue for each d.
The coefficients of Lorentzian polynomials satisfy a log-concavity inequality as in Proposition 2.4 below. It is in this sense that Lorentzian polynomials generalize the notion of log-concavity.
Proposition 2.4 ([BH19, Proposition 9.4]). If f (x) = α c α x α is a homogeneous polynomial on n variables so that N (f ) is Lorentzian, then for any α ∈ N n and any i, j ∈ [n] the inequality This proposition can be seen as a consequence of Cauchy's Interlacing Theorem. We recall below a special case of Cauchy's Interlacing Theorem, which we will use later. . Let A be a symmetric n × n matrix, and let S ⊆ [n], and m = |S|. Let B = A S be thde m × m principal submatrix of A given by B = (a ij ) i,j∈S . Let α 1 ≤ · · · ≤ α n be the eigenvalues of A and let β 1 ≤ · · · ≤ β m be the eigenvalues of B. Then for every j ∈ [m], α j ≤ β j ≤ α n−m+j . In other words, the jth smallest eigenvalue of A is at most the jth smallest eigenvalue of B, and the jth largest eigenvalue of A is at least the jth largest eigenvalue of B.
We recall two important theorems about Lorentzian polynomials here: Theorem 2.6 ([BH19, Theorem 2.10]). If f ∈ L d n is a Lorentzian polynomial in n variables, and A is an is a Lorentzian polynomial.

Flow polytopes.
Recall the definition of flow polytopes in (1.3). We record several properties of them here which we will be using in later sections.
Lemma 2.9 ( [Sch03]). For any graph G on the vertex set [n + 1], the vertices of the flow polytope F G (e 1 − e n+1 ) are unit flows with support equal to p, where p is an increasing path from vertex 1 to vertex n + 1.
Proposition 2.10 ([BV08, Section 3.4]). For nonnegative integers a 1 , . . . , a n and G a graph on the vertex set [n + 1] we have that Theorem 2.11 (Baldoni-Vergne volume formula, [BV08, Theorem 38]). Let G be a directed graph on the vertex set [n + 1] with a unique sink, so that edges are oriented from a smaller vertex to a larger vertex. Let The sum is over weak compositions j = (j 1 , . . . , j n ) of |E(G)| − n that dominate (out 1 , . . . , out n ), that is, for every i ∈ [n] we have j 1 + · · · + j i ≥ out 1 + · · · + out i .
In the above

Projections of flow polytopes onto generalized permutahedra
In this section, we define the projections ϕ : F G (a) → P(G; a) and ψ : F G (a) → Q(G; a), where P(G; a) and Q(G; a) are y-generalized permutahedra (see Propositions 3.4 and 3.6). We study their fibers in Section 4, leading us to explicit expressions for the polynomials σ ϕ G(a) and σ ψ G(a) ; see Corollary 4.3. In Section 5 we use these expressions to prove Theorem 5.1.
Notational Conventions for Sections 3, 4 and 5. Unless specified otherwise, G denotes a loopless directed graph on the vertex set [n + 1] vertices with a unique sink. Every edge of G is oriented from its smaller vertex to its larger vertex. All flow polytopes F G (a) have netflow vector a ∈ Z n ≥0 × Z ≤0 . For a finite set S, we denote by R S the real vector space consisting of R-linear combinations of elements in S; observe that for sets S ⊆ T , the vector space R S canonically embeds in R T as a coordinate hyperspace. We write R n to denote R [n] . Definition 3.2 (see Example 3.3). We denote by S G the set of all edges incident to the sink, that is, For i ∈ [n], let S G,i ⊆ S G be the set of edges incident to n + 1 which can be reached from vertex i, that is, if G denotes the transitive closure of G, then Denote by T G the set of all vertices incident to the sink, that is, For i ∈ [n], let T G,i ⊆ T G be the set of vertices adjacent to n + 1 which can be reached from vertex i, that is, Example 3.3. Let G be as in Figure 1. The set S G ⊆ E(G) consists of the blue edges, while S G,2 consists of the four blue edges emanating from vertices 2 and 4. If G denotes the graph obtained from G by removing the edge (2, 3; 1) ∈ E(G), then S G ,2 would only consist of the two blue edges emanating from vertex 2. Proposition 3.4. There is a projection where P(G; a) ⊆ R S G is the y-generalized permutahedron defined by The map ϕ is given by projecting a flow in F G (a) to the coordinates corresponding to edges in S G .
Proof. Proposition 2.10 asserts that Because linear maps factor through Minkowski sums, we obtain Observe that ϕ(F G (e i − e n+1 )) = ∆ S G,i , because their vertex sets coincide: Lemma 2.9 asserts that the vertices of F G (e i − e n+1 ) are unit flows on paths p from i to n + 1; under ϕ, the vertex of F G (e i − e n+1 ) corresponding to p is mapped to the vertex of ∆ S G,i corresponding to the (unique) edge in p that is incident to n + 1. The claim ϕ(F G (a)) = P(G; a) follows.
We note that a special case of Proposition 3.4 was considered in [MS17, Section 4].
For x ∈ F G (a), and ϕ as in Proposition 3.4, define Note that if i ∈ T G , or equivalently that if I i = ∅, then ef(x) i = 0. Thus, we may regard ef(x) as a vector in R T G (however, it will be useful to regard them as elements of R n whose coordinates indexed by [n] \ T G are zero).
Note also that ef(x) depends only on coordinates of Hence ef(ϕ(x)) def = ef(x), is well defined since ϕ leaves the coordinates of x corresponding to edges in S G unchanged.
Proposition 3.6. There is a projection The map ψ is given by sending x → ef(x). The map ψ factors through ϕ, that is, the following diagram commutes: Proof. As in the proof of Proposition 3.4, it will suffice to show ψ(F G (e i − e n+1 )) = ∆ T G,i . Lemma 2.9 asserts that that the vertices of F G (e i − e n+1 ) are unit flows on paths p from i to n + 1; under ψ, the vertex of F G (e i − e n+1 ) corresponding to p is mapped to the vertex of ∆ T G,i corresponding to the (unique) vertex t of G for which p contains an edge from t to n + 1. That the diagram commutes is the statement that ef(ϕ(x)) def = ef(x) is well defined, as discussed after Definition 3.5.

The fibers of ϕ and ψ
In order to study σ ϕ G(a) and σ ψ G(a) as defined in (1.4) and (1.5), we rewrite them as in equations (4.1) and (4.2) below; the validity of these equations follows from Propositions 3.4 and 3.6. Equations (4.1) and (4.2) make it evident that in order to explicitly compute the coefficients of the monomials appearing in σ ϕ G(a) and σ ψ G(a) (Corollary 4.3) we need to compute the fibers of ϕ and ψ, which is what we accomplish in Theorem 4.1 and Corollary 4.2, respectively.
For brevity of notation, we index the coordinates of a point x ∈ R S G with (i; k), which is shorthand for the edge (i, n + 1; k) ∈ S G . We define the polynomials Theorem 4.1. Given a point x ∈ P(G; a), the preimage S x def = ϕ −1 (x) ∩ F G (a) is a translation of the flow polytope F G (a 1 − ef(x) 1 , . . . , a n − ef(x) n , 0). For x ∈ Z S G , S x is integrally equivalent to F G (a 1 − ef(x) 1 , . . . , a n − ef(x) n , 0).
We emphasize that a = (a 1 , . . . , a n , − n i=1 a i ) with a i ≥ 0, and that for any x ∈ P(G; a) we have with the second equality by the fact that i I i = S G and the last equality by the definition of P(G; a).
Proof. Let ϕ ⊥ : R E(G) → R E(G) denote the projection sending components corresponding to edges in S G to zero. Note that ϕ and ϕ ⊥ project R E(G) to orthogonal complements, so ϕ ⊥ is necessarily an injection from S x onto its image (since points in S x are all mapped to x by ϕ). To clean up notation, we will write Restricting an a-flow in S x onto just the edges in G| [n] gives a (nonnegative) flow with netflow precisely a i − ef(x) i on vertex i. Hence, ϕ ⊥ is a map S x → F G (z 1 , . . . , z n , 0); furthermore, the inverse F G (z 1 , . . . , z n , 0) → S x is translation byx a 1 − x 1 , . . . , a n − x n , 0), since an a-flow in T x restricted onto just the edges in G| [n] gives a flow with netflow precisely a i − x i on vertex i. The fiber ψ −1 (p) ∩ F G (a) of any point p ∈ F G (a 1 − x 1 , . . . , a n − x n , 0) is equal to {p} × i∈T G x i ∆ Ii . The claim follows. K G (a 1 − ef(p) 1 , . . . , a n − ef(p) n ) Proof. The number of integer points of F G (a) is given by the Kostant partition function K G (a). Combining this fact with Theorem 4.1 and Corollary 4.2 gives the desired result.

Normalized projections of integer point transforms are Lorentzian
In this section, we show that N (σ ϕ G(a) ) and N (σ ψ G(a) ) are Lorentzian; see Theorem 5.1. In order to prove this we begin with a series of reductions (Proposition 5.8 and Lemma 5.11). Then, a combinatorial symmetry (Lemma 5.14) allows us to realize Hessians of repeated partial derivatives of σ ϕ G(a) as Hessians of repeated partial derivatives of volume polynomials.
We begin by formally stating the main result of this section.
As N (σ ϕ G(a) ) is Lorentzian, the coefficients of σ ϕ G(a) satisfy a log-concavity inequality (see Proposition 2.4), which is equivalent to: Corollary 5.2 (cf. [HMMS19, Proposition 11]). For any directed graph G on the vertex set [n] and for any v ∈ Z n we have: Note that Corollary 5.2 also follows from the classical Alexandrov-Fenchel inequalities for mixed volumes, since K G (v) can be seen as mixed volumes of Minkowski sums of flow polytopes.
A first stepping stone towards Theorem 5.1 is to reduce to the problem of showing N (σ ϕ G(a) ) is Lorentzian for all G; this is the content of Proposition 5.8. In order to do this we introduce the following construction.
Definition 5.3. For a graph G, we denote by G ex = (V ex , E ex ) the graph obtained from G by adding formal vertices i ex for each vertex i ∈ T G , by replacing edges (i, n + 1; j) ∈ S G with edges (i, i ex ; j), and by adding edges (i ex , n + 1; 1) for each i ex ∈ T G . Formally, we have See Figure 2 for an example. The graph G can be recovered from G ex by a series of contractions, so we call G ex the extension of G. Definition 5.4. For any two vectors p = (p 1 , . . . , p m ) ∈ R m and q = (q 1 , . . . , q n ) ∈ R n , we denote by p ⊕ q ∈ R m+n their concatenation, that is, p ⊕ q = (p 1 , . . . , p m , q 1 , . . . , q n ) ∈ R m+n .
For a netflow a for G satisfying the conventions of this paper, we denote by a ex the netflow for G ex given by a| [n] ⊕ 0 T G ex ⊕ −a n+1 = (a 1 , . . . , a n , 0, . . . , 0 note that a ex also satisfies the conventions of this paper. Lemma 5.5. The bijection T G ↔ S G ex given by i ↔ (i ex , n + 1; 1) induces an isomorphism on the real vector spaces R T G and R S G ex by renaming basis elements according to the bijection. This isomorphism restricts to an integral equivalence Q(G; a) ≡ P(G ex ; a ex ).
Proof. By definition, Since for every i ∈ T G ex we have a ex i = 0, and for every i ∈ [n] we have a ex i = a i , we may write P(G ex ; a ex ) = i∈ [n] a i ∆ S G ex ,i ; furthermore, we have S G ex ,i = {(i ex , n + 1; 1) : i ∈ T G,i }. Thus, the isomorphism sends ∆ T G,i to ∆ S G ex ,i ; passing to the Minkowski sum, we obtain the integral equivalence Q(G; a) ≡ P(G ex ; a ex ).
Lemma 5.6. The bijection E(G) ↔ E(G ex ) \ S G ex given by sending an edge (i, n + 1; k) ∈ S G to (i, i ex ; k) ∈ E(G ex ) \ S G ex and an edge (i, j; k) ∈ E(G) \ S G to (i, j; k) ∈ E(G ex ) \ S G ex induces an isomorphism on the real vector spaces spanned by E(G) and E(G ex ) \ S G ex by renaming basis elements according to the bijection. For every q ∈ Z T G , this isomorphism restricts to an integral equivalence In light of Corollary 4.2, note that the left side of Equation (5.1) is the fiber of q under F G (a) → Q(G; a). For brevity of notation, let us temporarily denote by q ∈ Z S G ex the image of q ∈ Z T G under the isomorphism in Lemma 5.5. In this notation, Theorem 4.1 implies that the right side of Equation (5.1) is (integrally equivalent to) the fiber of q ∈ Z S G ex under F ex G (a ex ) → P(G ex , a ex ). Proof of Lemma 5.6. A point f ∈ F G| [n] (a| [n] − q) × i∈T G q i ∆ Ii can be interpreted as a flow in F G (a) with outflow q i at each vertex i ∈ T G , by Corollary 4.2. Under the isomorphism in Lemma 5.6, f gets mapped to a flow in G ex | [n] T G ex with netflow a i at each vertex i ∈ [n] and netflow −q i at each vertex i ex ∈ T G ex . In other words, the image is in Conversely, the preimage of a flow Lemma 5.7. The bijection T G ↔ S G ex given by i ↔ (i ex , n+1; 1) induces an isomorphism on the polynomial rings R[(x i ) i∈T G ] and R[(x i ) i∈S G ex ] by renaming variables according to the bijection. Under this isomorphism, the polynomial N (σ ψ G(a) ) is sent to N (σ ϕ G ex (a ex ) ). Proof. Explicitly, we need to show that the polynomials By Lemma 5.6, the fibers ψ −1 (q) appearing in Equation (5.2) and the corresponding fibers ϕ −1 (p) appearing in Equation (5.3) are integrally equivalent. Hence the coefficients of the monomials appearing in Equations (5.2) and (5.3) match.
Proposition 5.8. Suppose N (σ ϕ G(a) ) is Lorentzian for every G. Then N (σ ψ G(a) ) is Lorentzian for every G. Proof. Lemma 5.7 asserts that up to renaming variables, we have the equality ). By assumption, N (σ ϕ G ex (a ex ) ) is Lorentzian.
has at most one positive eigenvalue. Then N (σ ϕ G(a) ) is Lorentzian. Proof. The support of N (σ ϕ G(a) ) is M-convex by Proposition 3.4. By Corollary 4.3, the ij-th element of K d is the coefficient of x d+ei+ej in σ ϕ G(a) ; equivalently, K d is the Hessian of ∂ d N (σ ϕ G(a) ). Since, by assumption, K d has at most one positive eigenvalue, Lemma 2.3 asserts that N (σ ϕ G(a) ) is Lorentzian. In other words, G − is obtained from G by replacing, for each i ∈ [n] with M (i, n + 1) ≥ 1, the set of edges connecting i to the sink with a single edge connecting i to the sink. See Figure 3 for an example. Note that since G − has at most one edge connecting i to n + 1 for any i, we have 3 4 5 Figure 3. The graph G from Figures 1 and 2. The graph G − constructed from G is shown beside it; see Definition 5.10.
Proof. The support of N (σ ϕ G(a) ) is M-convex by Proposition 3.4. By Lemma 5.9, we need to show that for every d ∈ ( has at most one positive eigenvalue. The matrix K d is obtained from the |T G | × |T G | matrix first by repeating the ith row M (i, n + 1) many times for each i, and then by repeating the ith column M (i, n + 1) many times for each i. Note that the rank of K − ef(d) is equal to the rank of K d ; we write Observe, by Corollary 4.3 that the ij-th entry of By assumption, N (σ ϕ G − (a) ) is Lorentzian; hence, Lemma 2.3 asserts that K − ef(d) has at most one positive eigenvalue for any ef(d) ∈ ( n i=1 a i − 2)∆ T G ∩ Z T G . In particular it has at least r − 1 negative eigenvalues. Note also that K − ef(d) is a principal submatrix of K d ; by Cauchy's Interlacing Theorem (Proposition 2.5), the eigenvalues α 1 ≤ α 2 ≤ · · · ≤ α |S G | of K d and the eigenvalues β 1 ≤ · · · ≤ β |T G | of K − ef(d) satisfy Since K − ef(d) has at least r − 1 negative eigenvalues, so K d also has at least r − 1 negative eigenvalues. Furthermore, K d has rank r. Hence, K d also has at most one positive eigenvalue, and (N (σ ϕ G(a) ))(x i;k ) is Lorentzian.
Definition 5.13. For a graph G as in the conventions of this section, denote by G r the graph obtained by "flipping" G| [n] , that is, V (G r ) = [n] and (i, j) ∈ E(G r ) ⇐⇒ (n + 1 − j, n + 1 − i) ∈ E(G| [n] ).
Equivalently, G r is obtained by relabeling the vertices of G| [n] by the map i → n + 1 − i and reversing the orientation of edges. See Figure 4 for an example.
The symmetry between G| [n] and G r underpins the following lemma, crucial for the proof of Theorem 5.1: Definition 5.15. Let G − be a graph satisfying the conventions of this section as well as M (i, n + 1) ≤ 1 for every i, see Definition 5.10. Denote by P T the permutation matrix corresponding to the order-reversing permutation i → |T G | + 1 − i; this is the matrix consisting of 1's on the antidiagonal and 0 everywhere else.
Proof. Note that the i, j-th entry , where the last equality is an application of Lemma 5.14.
The final piece required to prove Theorem 5.1 is the existence of a quadratic Lorentzian polynomial whose Hessian is K − ef(d) . We are ready to accomplish this now: where outd i denotes the outdegree of G at vertex i. Note that z + o ≥ 0, since for i ≤ n we have and for i ≥ n + 1 we have z i = 0. The Baldoni-Vergne formulas, Theorem 2.11, applied to G says that By Theorem 2.7, vol F G ( x) is Lorentzian. Hence, so is where the equality is an application of Lemma 2.2. Let A be the N × N diagonal matrix whose ith diagonal entry is 1 if i ∈ {n + 1 − j : j ∈ T G } and 0 otherwise; by Theorem 2.6 applied to f = ∂ z+ o vol F G ( x) and A as above, the quadratic polynomial is Lorentzian and its Hessian has at most one positive eigenvalue. The rows and columns of this Hessian are naturally indexed by {n + 1 − j : j ∈ T G }, and its i, j-th entry is the coefficient of . This coefficient is K G r ( z + e i + e j ). By Proposition 5.17, its Hessian is precisely K − ef(d) . We have thus shown that K − ef(d) has at most one positive eigenvalue, completing the proof.

On projections of polytopes in general
Recall the question stemming from Theorem 5.1, as well as other examples mentioned in the Introduction: Question 1.1. What conditions on the polytope/projection pair ascertain that the normalization of the projection of the integer point transform of the polytope is Lorentzian?
Note that ϕ is a projection onto a coordinate hypersurface and the flow polytope F G (a) we are projecting lives in the nonnegative orthant. It is worth noting that once we have a Lorentzian polynomial f which equals the normalized projection onto a coordinate hypersurface of an integer point transform of a polytope which belongs to the nonnegative orthant, then any derivative of f is (1) Lorentzian, (2) also the normalized projection onto a coordinate hypersurface of an integer point transform of a polytope which belongs to the nonnegative orthant. We formalize this observation here. we also require that ϕ is a projection onto a coordinate n-dimensional hypersurface. Without loss of generality, we may assume ϕ is projection onto the first n components.
Observe that ϕ(P ) ⊆ H n + lives inside the nonnegative orthant of R n and also has integral vertices. To an admissible pair, we associate a polynomial σ ϕ P obtained by projecting the integer point transform of P according to ϕ; specifically, where x = (x 1 , . . . , x n ), and ϕ −1 (p) is interpreted as a subset of P . (Note that ϕ(P ) ⊆ H n + implies σ ϕ P is actually a polynomial.) Proposition 6.2. Let f (x 1 , . . . , x n ) be a Lorentzian polynomial so that f = N (σ ϕ P ) for some admissible pair (P, ϕ). c α x α−ei .
Since e i ∈ R n = im ϕ, we have ϕ(P i ) = ϕ(P ∩ H m +i ) + {−e i } = ϕ(P ) ∩ H n +i + {−e i }. A point β ∈ ϕ(P i ) if and only if α def = β + e i ∈ ϕ(P ) ∩ H n +i . Furthermore, the fiber ϕ −1 (β) ∩ P i is equal, up to translation by e i , to the fiber ϕ −1 (α) ∩ P . Thus Comparing the above expression to the definition of σ ϕ P , we have verified Equation (6.1) holds. Remark 6.3. We emphasize that the pair (P i , ϕ) is admissible when (P, ϕ) is admissible. Furthermore, as discussed in the proof of Proposition 6.2, ϕ(P i ) = ϕ(P ) ∩ H n +i + {−e i }.
We conclude by another intriguing question stemming from our work: which Lorentzian polynomials arise naturally as normalized projections of integer point transforms of polytopes?