Critical points of discrete periodic operators

We study the spectra of operators on periodic graphs using methods from combinatorial algebraic geometry. Our main result is a bound on the number of complex critical points of the Bloch variety, together with an effective criterion for when this bound is attained. We show that this criterion holds for Z^2- and Z^3-periodic graphs with sufficiently many edges and use our results to establish the spectral edges conjecture for some Z^2-periodic graphs.


Introduction
The spectrum of a Z d -periodic self-adjoint discrete operator L consists of intervals in R. Floquet theory reveals that the spectrum is the image of the coordinate projection to R of the Bloch variety (also known as the dispersion relation), an algebraic hypersurface in (S 1 ) d × R.This coordinate projection defines a function λ on the Bloch variety, which is our main object of study.
When the operator is discrete, the complexification of the Bloch variety is an algebraic variety in (C × ) d × C. Thus techniques from algebraic geometry and related areas may be used to address some questions in spectral theory.In the 1990's Gieseker, Knörrer, and Trubowitz [16] used algebraic geometry to study the Schrödinger operator on the square lattice Z 2 with a periodic potential and established a number of results, including Floquet isospectrality, the irreducibility of its Fermi varieties, and determined the density of states.Recently there has been a surge of interest in using algebraic methods in spectral theory.This includes investigating the irreducibility of Bloch and Fermi varieties [11,12,23,25], Fermi isospectrality [26], density of states [20], and extrema and critical points of the projection λ on Bloch varieties [1,9,25].We use techniques from combinatorial algebraic geometry and geometric combinatorics [31] to study critical points of the function λ on the Bloch variety of a discrete periodic operator.We now discuss motivation and sketch our results.Some background on spectral theory is sketched in Section 1, and Section 2.1 gives some background from algebraic geometry.
An old and widely believed conjecture in mathematical physics concerns the structure of the Bloch variety near the edges of the spectral bands.Namely, that for a sufficiently general operator L (as defined in Section 1.1), the extrema of the band functions λ j on the Bloch variety are nondegenerate in that their Hessians are nondegenerate quadratic forms.This spectral edges nondegeneracy conjecture is stated in [22,Conj. 5.25], and it also appears in [5,21,27,28].Important notions, such as effective mass in solid state physics, the Liouville property, Green's function asymptotics, Anderson localization, homogenization, and many other assumed properties in physics, depend upon this conjecture.
The spectral edges conjecture states that for generic parameters, each extreme value is attained by a single band, the extrema are isolated, and the extrema are nondegenerate.We discuss progress for discrete operators on periodic graphs.In 2000, Klopp and Ralston [18] proved that for Laplacians with generic potential each extreme value is attained by a single band.In 2015, Filonov and Kachkovskiy [13] gave a class of two-dimensional operators for which the extrema are isolated.They also show [13,Sect. 6] that the spectral edges conjecture may fail for a Laplacian with general potential, which does not have generic parameters in the sense of Section 1.1.Most recently, Liu [25] proved that the extrema are isolated for the Schrödinger operator acting on the square lattice.
We consider a property which implies the spectral edges nondegeneracy conjecture: A family of operators has the critical points property if for almost all operators in the family, all critical points of the function λ (not just the extrema) are nondegenerate.Algebraic geometry was used in [9] to prove the following dichotomy: For a given algebraic family of discrete periodic operators, either the critical points property holds for that family, or almost all operators in the family have Bloch varieties with degenerate critical points.
In [9], this dichotomy was used to establish the critical points property for the family of Laplace-Beltrami difference operators on the Z 2 -periodic diatomic graph of Figure 1.Bloch varieties for these operators were shown to have at most 32 critical points.A single example was computed to have 32 nondegenerate critical points.Standard arguments from algebraic geometry (see Section 5) implied that, for this family, the critical points property, and therefore also the spectral edges nondegeneracy conjecture, holds.
We extend part of that argument to operators on many periodic graphs.Let L be a discrete operator on a Z d -periodic graph Γ (see Section 1).Its (complexified) Bloch variety is a hypersurface in the product (C × ) d × C of a complex torus and the complex line defined by a Laurent polynomial D(z, λ).The last coordinate λ, corresponding to projection onto the spectral axis, is the function on the Bloch variety whose critical points we study.Accordingly, we will call critical points of the function λ on the Bloch variety "critical points of the Bloch variety."One contribution of this paper is to shift focus from spectral band functions λ j defined on a compact torus to a global function on the complex Bloch variety.Another is to use the perspective of nonlinear optimization to address a question concerning the spectrum of a discrete periodic operator.
We state our first result.Let Γ be a connected Z d -periodic graph (as in Section 1.1).Fix a fundamental domain W for the Z d -action on the vertices of Γ.The support A(Γ) of Γ records the local connectivity between translates of the fundamental domain.It is the set of a ∈ Z d such that Γ has an edge with endpoints in both W and a+W .This bound uses an outer approximation for the Newton polytope of D(z, λ) (see Lemma 4.1) and a study of the equations defining critical points of the function λ on the Bloch variety, called the critical point equations (8).Corollary 2.5 is a strengthening of Theorem A. When the bound is attained all critical points are isolated.
Example.We illustrate Theorem A on the example from [9,Sect. 4]. Figure 1 shows a periodic graph Γ with d = 2 whose fundamental domain W has two vertices and its support A(Γ) consists of the columns of the matrix ( 0 1 0 −1 0 0 0 1 0 −1 ). Figure 1 also displays the convex hull of A(Γ).As |W | = 2 and vol(A(Γ)) = 2, Theorem A implies that any Bloch variety for an operator on Γ has at most critical points, which is the bound demonstrated in [9].⋄ The bound of Corollary 2.5 arises as follows.There is a natural compactification of (C × ) d ×C by a projective toric variety X associated to the Newton polytope, P, of D(z, λ) [15,Ch. 5].The critical point equations become linear equations on X whose number of solutions is the degree of X.By Kushnirenko's Theorem [19], this degree is the normalized volume of P , (d+1)!vol(P ).This bound is attained exactly when there are no solutions at infinity, which is the set ∂X := X (C × ) d × C of points added in the compactification.The compactified Bloch variety is a hypersurface in X.A vertical face of P is one that contains a segment parallel to the λ-axis.Corollary 3.9 shows that when P has no vertical faces, any solution on ∂X to the critical point equations is a singular point of the intersection of this hypersurface with ∂X.We state a simplified version of Corollary 3.9.
Theorem B. If P has no vertical faces, then the bound of Corollary 2.5 is attained exactly when the compactified Bloch variety is smooth along ∂X.
We give a class of graphs whose typical Bloch variety is smooth at infinity and whose Newton polytopes have no vertical faces.A periodic graph Γ is dense if it has every possible edge, given its support A(Γ) and fundamental domain W (see Section 4).The following is a consequence of Corollary 3.9 and Theorem 4.2.
Theorem C. When d = 2 or 3 the Bloch variety of a generic operator on a dense periodic graph is smooth along ∂X, its Newton polytope has no vertical faces, and the bound of Theorem A is attained.Theorem C is an example of a recent trend in applications of algebraic geometry in which a highly structured optimization problem is shown to unexpectedly achieve a combinatorial bound on the number of critical points.A first instance was [9], which inspired [4] and [24].
Section 1 presents background on the spectrum of an operator on a periodic graph, and formulates our goal to bound the number of critical points of the function λ on the Bloch variety.At the beginning of Section 3, we recast extrema of the spectral band functions using the language of constrained optimization.Theorems A, B, and C are proven in Sections 2, 3, and 4. In Section 5, we use these results to prove the spectral edges conjecture for operators on three periodic graphs.

Operators on periodic graphs
Let d be a positive integer.We write C × := C {0} for the multiplicative group of nonzero complex numbers and T := {z ∈ C × | |z| = 1} for its maximal compact subgroup.Note that if z ∈ T, then z = z −1 .We write edges of a graph as pairs, (u, v) with u, v vertices, and understand that (u, v) = (v, u).
1.1.Operators on periodic graphs.For more, see [2,Ch. 4].A (Z d -)periodic graph is a simple (no multiple edges or loops) connected undirected graph Γ with a free cocompact action of Z d .Thus Z d acts freely on the vertices, V(Γ), and edges, E(Γ), of Γ preserving incidences, and Z d has finitely many orbits on each of V(Γ) and E(Γ).Z 2 -periodic graphs.One is the honeycomb lattice and the other is an abelian cover of K 4 , the complete graph on four vertices.
It is useful but not necessary to consider Γ immersed in R d so that Z d acts on Γ via translations.The graphs in Figure 2 are each immersed in R 2 , and for each we show two independent vectors that generate the Z 2 -action.
Choose a fundamental domain for this Z d -action whose boundary does not contain a vertex of Γ.In Figure 2, we have shaded the fundamental domains.Let W be the vertices of Γ lying in the fundamental domain.Then W is a set of representatives of Z d -orbits of V(Γ).Every Z d -orbit of edges contains one or two edges incident on vertices in W .An edge incident on W has the form (u, a+v) for some u, v ∈ W and a ∈ Z d .(If a = 0, then u = v as Γ has no loops, and there are no restrictions when a = 0.) The support A(Γ) of Γ is the set of a ∈ Z d such that (u, a+v) ∈ E(Γ) for some u, v ∈ W .This finite set depends on the choice of fundamental domain and it is centrally symmetric in that A(Γ) = −A(Γ).As Γ is connected, the Z-span of A(Γ) is Z d .For both graphs in Figure 2, this set consists of the columns of the matrix ( 0 1 0 −1 0 0 0 1 0 −1 ).A labeling of Γ is a pair of functions e : E(Γ) → R (edge weights) and V : V(Γ) → R (potential) that is Z d -invariant (constant on orbits).The set of labelings is the finitedimensional vector space R E × R W , where E is the set of orbits on E(Γ).Given a labeling c = (e, V ), we have the discrete operator L c acting on functions f on V(Γ).Then L c (f ) is defined by its value at u ∈ V(Γ), We call L c a discrete periodic operator on Γ, and may often omit the subscript c.It is a bounded self-adjoint operator on the Hilbert space ℓ 2 (Γ) of square-summable functions on V(Γ), and has real spectrum.1.2.Floquet theory.As the action of Z d on Γ commutes with the operator L, we may apply the Floquet transform, which reveals important structure of its spectrum.References for this Floquet theory include [2,21,22].
The Floquet (Fourier) transform is a linear isometry ℓ 2 (Γ) to square-integrable functions on T d , the compact torus, with values in the vector space C W .The torus T d is the group of unitary characters of Z d .For z ∈ T d and a ∈ Z d , the corresponding character value is the Laurent monomial Thus f is determined by its values at the vertices W in the fundamental domain.Let f ∈ L 2 (T d , C W ). Then for u ∈ W , f (u) is a function on T d .The action of the operator L on the Floquet transform f is given by the formula as f (a+v) = z a f (v).The exponents a which appear lie in the support A(Γ) of Γ.The simplicity of this expression is because L commutes with the Z d -action.Thus in the standard basis for C W , the operator L becomes multiplication by a square matrix whose rows and columns are indexed by elements of W . Writing δ u,v for the Kronecker delta function, the matrix entry in position (u, v) is the function e (u,a+v) z a .
Example 1.1.Let Γ be the hexagonal lattice from Figure 2. Figure 3 shows a labeling in a neighborhood of its fundamental domain.Thus W = {u, v} consists of two vertices and there are three (orbits of) edges, with labels α, β, γ.Let (x, y) ∈ T 2 .The operator L is Collecting coefficients of f (u), f (v), we represent L by the 2 × 2-matrix, whose entries are Laurent polynomials in x, y.Notice that the support A(Γ) of Γ equals the set of exponents of monomials which appear in L. Observe that for (x, y) ∈ T 2 , L T = L, so that L is Hermitian, showing again that the operator L is self-adjoint.⋄ What we saw in Example 1.1 holds in general.In the standard basis for C W , L = L c is multiplication by a |W | × |W |-matrix L(z) = L c (z) with each entry (3) a finite sum of monomials with exponents from A(Γ) (a Laurent polynomial with support A(Γ)).Note that (u, a+v) ∈ E(Γ) if and only if (−a+u, v) ∈ E(Γ), these edges have the same label, and for z ∈ T d , z a = z −a .Thus for z ∈ T d , the matrix is Hermitian, as L(z) T = L(z −1 ) = L(z).
1.3.Critical points of the Bloch variety.As L(z) is Hermitian for z ∈ T d , its spectrum is real and consists of its |W | eigenvalues These eigenvalues vary continuously with z ∈ T d , and λ j (z) is called the jth spectral band function, λ j : T d → R. Its image is an interval in R, called the jth spectral band.The eigenvalues (5) are the roots of the characteristic polynomial which we call the dispersion function.Its vanishing defines a hypersurface ( 7) called the Bloch variety of the operator L1 .The Bloch variety is the union of |W | branches with the jth branch equal to the graph of the jth spectral band function.The image of the Bloch variety under the projection to R is the spectrum σ(L) of the operator L. This projection is a function λ on the Bloch variety.Identifying the jth branch/graph with T d , the restriction of λ to that branch gives the corresponding spectral band function λ j .Figure 4 shows this for the operator L on the hexagonal lattice with edge weights 6, 3, 2 and zero potential V -for this we unfurl T 2 , representing it by [− π 2 , 3π 2 ] 2 ⊂ R 2 , which is a fundamental domain in its universal cover.(That is, by quasimomenta in [− π 2 , 3π 2 ] 2 .)It has two branches with each the graph of the corresponding spectral band function.An endpoint of a spectral band (spectral edge) is the image of an extremum of some band function λ j (z).
For the hexagonal lattice at these parameters, each band function has two nondegenerate extrema, and these give the four spectral edges.These are also local extrema of the function λ on the Bloch variety.
The spectral edges conjecture [22,Conj. 5.25] for a periodic graph Γ asserts that for generic values of the parameters (e, V ), each spectral edge is attained by a single band, the extrema on the Bloch variety are isolated, and all extrema are nondegenerate (the spectral band function λ j has a full rank Hessian matrix).Here, generic means that there is a nonconstant polynomial p(e, V ) in the parameters such that when p(e, V ) = 0, these desired properties hold.
The entries in the matrix L(z) and the function ( 6) defining the Bloch variety are all (Laurent) polynomials.In this setting it is natural to allow complex parameters, e : (6).
In passing to the complex Bloch variety we may no longer distinguish branches λ j (z) of λ.At a smooth point (z 0 , λ 0 ) whose projection z to (C × ) d is regular (in that ∂D ∂λ (z 0 , λ 0 ) = 0), there is a locally defined function f of z with λ 0 = f (z 0 ) and D(z, f (z)) = 0 on its domain, but this is not necessarily a global function of z.Consequently, we will consider the projection to the last coordinate to be a function λ on the Bloch variety, and then study its differential geometry, including its critical points.
Nondegeneracy of spectral edges is implied by the stronger condition that all critical points of the function λ on the complex Bloch variety are nondegenerate.Understanding the critical points of λ is a first step.Our aim is to bound the number of (isolated) critical points of λ on the Bloch variety of a given operator L, give criteria for when the bound is attained, prove that it is attained for generic operators on a class of graphs, and finally to use these results to prove the spectral edges conjecture for 2 19 + 2 graphs.We treat these in the following four sections.

Bounding the number of critical points
We first recast extrema of spectral band functions in terms of constrained optimization.The complex Bloch variety is the hypersurface Var(D(z, λ)) in (C × ) d × C defined by the vanishing of the dispersion function D(z, λ).Critical points of the function λ on the Bloch variety are points of the Bloch variety where the gradients in
) is an extreme value of the spectral band function λ j is also a critical point of the Bloch variety.Indeed, either the gradient ∇D vanishes at (z 0 , λ 0 ) or it does not vanish.If ∇D(z 0 , λ 0 ) = 0, then (z 0 , λ 0 ) is a critical point.If ∇D(z 0 , λ 0 ) = 0, then the Bloch variety is smooth at (z 0 , λ 0 ) and thus is a smooth point of the graph of λ j .As λ 0 = λ j (z 0 ) is an extreme value of λ j , the tangent plane is horizontal at (z 0 , λ 0 ).This implies that λ j is differentiable (by the implicit function theorem) and that ∂λ j ∂z i (z 0 , λ 0 ) = 0 for i = 1, . . ., d.Thus the gradients of λ and D at (z 0 , λ 0 ) are linearly dependent, showing that it is a critical point.⋄ Bézout's Theorem [29,Sect. 4.2.1]gives an upper bound on the number of isolated critical points: We may multiply each Laurent polynomial in (8) by a monomial to clear denominators and obtain ordinary polynomials.The product of their degrees is an upper bound for the number of the common zeroes that are isolated in the complex domain.Polyhedral bounds that exploit the structure of the Laurent polynomials are typically much smaller.Sources for these are [6,Ch. 7], [15,Ch. 5], and [30,Ch. 3].These results bound the number of isolated common zeroes, counted with multiplicities.An isolated common zero z 0 of polynomials f 1 , . . ., f d+1 on (C × ) d × C has multiplicity 1 exactly when the gradient of f 1 , . . ., f d+1 spans the cotangent space at z 0 ; otherwise its multiplicity exceeds 1 (see [6,Ch. 4 Def.2.1] and [7,Ch. 8.7 Def. 8]).
Let C[z ± , λ] be the ring of Laurent polynomials in z 1 , . . ., z d , λ where λ occurs with only nonnegative exponents.Note that is the set of exponents of monomials in ψ.The Newton polytope N (ψ) := conv(A(ψ)) of ψ is the convex hull of its support.Write vol(N (ψ)) for the (d+1)dimensional Euclidean volume of the Newton polytope of ψ.
Example 2.3.We continue the example of the hexagonal lattice.Writing ℓ for α+β+γ−λ, the dispersion function D(x, y; λ) of the matrix ( 4) is In Figure 5 the monomials in D(x, y; λ) label the columns of a 3 × 9 array which are their exponent vectors.Figure 5 also shows its Newton polytope, which has volume 2.
. Support and Newton polytope of the hexagonal lattice operator.
Theorem 2.4.For a polynomial ψ ∈ C[z ± , λ], the critical point equations for ψ When the bound is attained, all solutions are isolated.
We prove this at the end of the section.
As the Bloch variety is defined by the dispersion function D(z, λ) = det(L(z) − λI), we deduce the following from Theorem 2.4.
Corollary 2.5.The number of isolated critical points of the function λ on the Bloch variety for an operator L on a discrete periodic graph is at most (d+1)!vol(N (D)).
Theorem A follows from this and Lemma 4.1, which asserts that where e = (0, . . ., 0, 1).This containment implies the inequality We prove Theorem 2.4 and Corollary 2.5 after developing some preliminary results.

2.1.
A little algebraic geometry.For more from algebraic geometry, see [7,29].An (affine) variety is the set of common zeroes of some polynomials We also call this the set of solutions to the system f 1 = • • • = f r = 0. We may replace any factor C in C n by C × , and then allow the corresponding variable to have negative exponents.The complement of a variety X is a (Zariski) open set.This defines the Zariski topology in which varieties are the closed sets.A variety is irreducible if it is not the union of two proper subvarieties.For an irreducible variety, any nonempty open set is dense (even in the classical topology) and any nonempty classically open set is dense in the Zariski topology.Maps f : X → Y ⊂ C m of varieties are given by m polynomials on X and the image f (X) contains an open subset of its closure.
Suppose that X = Var(f 1 , . . ., f r ).The smooth (nonsingular) locus of X is the open subset of points of X where the Jacobian of f 1 . . ., f r has maximal rank on X.Let f be a single polynomial.A point x is a smooth point on the hypersurface Var (f ) defined by f if f (x) = 0, so that x ∈ Var (f ) and if the gradient ∇f (x) = ( ∂f ∂x 1 (x), . . ., ∂f ∂xn (x)) is nonzero, so that some partial derivative of f does not vanish at x.The point x ∈ Var (f ) is singular if all partial derivatives of f vanish at x.The kernel of the Jacobian at x ∈ X = Var(f 1 , . . ., f r ) is the (Zariski) tangent space at x.The dimension of an irreducible variety is the dimension of a tangent space at any smooth point.An isolated point x of X has multiplicity one exactly when it is nonsingular.
Remark 2.6.Our definition of smooth and singular points of a variety depends upon its defining polynomials.For example, the variety defined by (z −λ) 2 is singular at every point.This scheme-theoretic notion of singularity is essential to our arguments in Sections 3 and 4, and is standard in algebraic geometry.⋄ If X is irreducible, then any proper subvariety has smaller dimension.
We also have Bertini's Theorem: if X is smooth, then U may be chosen so that for every y ∈ U, f −1 (y) is smooth.
Projective space P(C n ) is the set of one-dimensional linear subspaces (lines) of C n and is compact.It has dimension n−1 and subvarieties are given by homogeneous polynomials.The set U 0 of lines spanned by vectors whose initial coordinate is nonzero is isomorphic to 2.2.Polyhedral bounds.The expression (d+1)!vol(N (ψ)) of Theorem 2.4 is the normalized volume of N (ψ).This is Kushnirenko's bound [15, Ch. 6, Thm.2.2] for the number of isolated solutions in (C × ) d+1 to a system of d+1 polynomial equations, all with Newton polytope N (ψ).To prove Theorem 2.4, we first explain why Kushnirenko's bound applies to the system (9), and then why it bounds the number of isolated solutions on the larger space (C × ) d × C.
For a monomial z a λ j in C[z ± , λ], a ∈ Z d and j ∈ N.For each i = 1, . . ., d, this monomial is an eigenvector for the operator A refined version of Kushnirenko's Theorem in which the polynomials may have different Newton polytopes is Bernstein's theorem [6, Sect.7.5], which is in terms of a quantity called mixed volume, whose properties are developed in [10, Ch.IV].The mixed volume of polytopes is monotone under inclusion of polytopes and it equals the normalized volume when all polytopes coincide.It follows that the theorems of Bernstein and Kushnirenko together give the bound of (d+1)!vol(N (ψ)) for the number of isolated solutions to the system (9) in (C × ) d+1 .To extend this to solutions in the larger space (C × ) d × C, we develop some theory of projective toric varieties.
2.3.Projective toric varieties.For Kushnirenko's Theorem and our extension, we replace the nonlinear equations ( 9) on (C × ) d × C by linear equations on a projective variety.We follow the discussion of [30,Ch. 3].Let f ∈ C[z ± , λ] be a polynomial with support A = A(f ).To simplify the presentation, we will at times assume that the origin 0 lies in A. The results hold without this assumption, as explained in [30,Ch. 3].
Writing C A for the vector space with basis indexed by elements of A, consider the map This map linearizes nonlinear polynomials.Indeed, write f as a sum of monomials, If {x (a,j) | (a, j) ∈ A} are variables (coordinate functions) on C A , then is a linear form on C A , and we have f ).Since 0 ∈ A, the corresponding coordinate x 0 of ϕ A is 1 and so the image of ϕ A lies in the principal affine open subset U 0 of the projective space P A := P(C A ) = P |A|−1 .This is the subset of P A where x 0 = 0 and it is isomorphic to the affine space C |A|−1 .We define X A to be the closure of the image ϕ A ((C × ) d+1 ) in the projective space P A , which is a projective toric variety.Because the map ϕ A is continuous on ( The map ϕ A is not necessarily injective; we describe its fibers.Let ZA ⊂ Z d+1 be sublattice generated by all differences α−β for α, β ∈ A. When 0 ∈ A this is the sublattice generated by A, and it has full rank d + 1 if and only if conv(A) has full dimension d + 1.
Let G A be Hom(Z d+1 /ZA, C × ) ⊂ (C × ) d+1 , which acts on (C × ) d × C. The fibers of ϕ A are exactly the orbits of G A on (C × ) d × C. If conv(A) does not have full dimension, then G A has positive dimension as do all fibers of ϕ A , otherwise G A is a finite group and ϕ A has finite fibers.On the torus (C × ) d+1 , G A acts freely and ϕ A ((C × ) d+1 ) is identified with , note that (C × ) d+1 acts on this through the homomorphism π that sends its last (λ) coordinate to {1}.Thus the fibers of ϕ A on (C × ) d × {0} are exactly the orbits of π(G A ) ⊂ (C × ) d .
Proposition 2.7.The dimension of X A is the dimension of conv(A).The fibers of ϕ A on (C × ) d+1 are the orbits of G A and its fibers on (C × ) d × {0} are the orbits of π(G A ).
We return to the situation of Theorem 2.4.Let ψ ∈ C[z ± , λ] be a polynomial with support A. As each polynomial in (9) has support a subset of A, each corresponds to a linear form on P A as in (10).The corresponding system of linear forms defines a linear subspace M ψ of P A .We have the following proposition (a version of [30,Lemma 3.5]).
Proposition 2.8.The solutions to (9) are the inverse images under ϕ A of points in the linear section ϕ Proof of Theorem 2.4.When vol(N (ψ)) = 0, so that N (ψ) does not have full dimension d + 1, then each fiber of ϕ A is positive-dimensional and so by Proposition 2.8 there are no isolated solutions to (9).
Suppose that vol(N (ψ)) > 0. Then every fiber of ϕ A is an orbit of the finite group G A .Over points of ϕ A ((C × ) d+1 ), each fiber consists of |G A | points and over ϕ A ((C × ) d ×{0} each fiber consists of |π(G A )| ≤ |G A | points.As X A is the closure of ϕ A ((C × ) d × C), the number of isolated points in X A ∩M ψ is at least the number of isolated points in ϕ A ((C × ) d ×C)∩M ψ , both counted with multiplicity.The degree of the projective variety X A is an upper bound for the number of isolated points in X A ∩ M ψ , which is explained in [30,Ch. 3.3].There, the product of |G A | and the degree of X A is shown to be (d+1)!vol(N(ψ)), the normalized volume of the Newton polytope of ψ.This gives the bound of Theorem 2.4.That all points are isolated when the bound of the degree is attained is Proposition 3.2 in the next section.

Proof of Theorem B
We give conditions for when the upper bound of Corollary 2.5 is attained.By Proposition 2.1, the critical points of the function λ on the Bloch variety Var (D) are the solutions in (C × ) d × C to the critical point equations (8).Let A = A(D) be the support of the polynomial D. The critical points are ϕ ) and M D is the subspace of P A defined by linear forms corresponding (as in (10)) to the polynomials in (8).For the bound of Theorem 2.4 and Corollary 2.5, note that the number of isolated points of X A ∩ M D is at most the product of the degree of X A with the cardinality of a fiber of ϕ A , which is (d+1)!vol(N (D)).We establish Theorem B concerning the sharpness of this bound by characterizing when the inequality of Theorem 2.4 is strict and then interpreting that for the critical point equations.
Remark 3.1.Let X ⊂ P n be a variety of dimension d and M ⊂ P n a linear subspace of codimension d.The number of points in X ∩M does not depend on M when the intersection is transverse; it is the degree of X [29, p. 234].When the intersection is not transverse, intersection theory gives a refinement [14,Ch. 6].For each irreducible component Z of the intersection X ∩ M, there is a positive integer-the intersection multiplicity along Z-such that the sum of these multiplicities is the degree of X.When Z is positive-dimensional this number is the degree of a zero-cycle constructed on Z (it is at least the degree of Z) and when Z is zero-dimensional (a point), it is the local multiplicity [29,Ch. 4].⋄ A consequence of Remark 3.1 is the following.
Proposition 3.2.Let X, M be as in Remark 3.1.The number (counted with multiplicity) of isolated points of X ∩ M is strictly less than the degree of X if and only if the intersection has a positive-dimensional component.
) for the image of ϕ A and ∂X A := X A X • A , the points of X A added to X • A when taking the closure.This is the boundary of X A .In the Introduction, points of ∂X A were referred to as 'lying at infinity'.
Proof.The inequality of Theorem 2.4 is strict if either of the following hold.
(1) X A ∩ M ψ has an isolated point not lying in X • A .(2) X A ∩ M ψ contains a positive-dimensional component Z.In (1), X A ∩ M ψ has isolated points in ∂X A ∩ M ψ , so the intersection is nonempty.In (2), Z is a projective variety of dimension at least one.The set X • A is an affine variety, and we cannot have Z ⊂ X • A as the only projective varieties that are also subvarieties of an affine variety are points.Thus Z ∩ ∂X A = ∅, which completes the proof.
3.1.Facial systems.We return to the general case of a toric variety.Let A ⊂ Z n be a finite set of points with corresponding projective toric variety X A ⊂ P A .We have the following description of the points of its boundary, X A ϕ A ((C × ) n ).
Let P := conv(A), the convex hull of A. The dot product with a nonzero vector minimizers is the face of P exposed by w.We have that F = conv(F ∩ A), and may write F for F ∩ A. As A ⊂ Z n , we only need integer vectors w ∈ Z n to expose all faces of P .If dim F = dim P − 1, then F is a facet.
For each face F of P , there is a corresponding coordinate subspace P F of P A -this is the set of points z = [z a | a ∈ A] ∈ P A such that a ∈ F implies that z a = 0.The image of the map ϕ F : (C × ) n → P F ⊂ P A has closure the toric variety X F .Its dimension is equal to the dimension of the face F .Write X • F for the image of ϕ F .This description and the following proposition is essentially [15, Prop.5.1.9].Proposition 3.4.The boundary of the toric variety X A is the disjoint union of the sets X • F for all the proper faces F of conv(A).
Let f = a∈A c a x a be a polynomial with support A. We observed that if Λ is the corresponding linear form (10) on P A , then the variety Var (f ) ⊂ (C × ) n of f is the pullback along ϕ A of X • A ∩ M, where M := Var (Λ) is the hyperplane defined by Λ.Let F be a proper face of P .Then X • F ∩ M pulls back along ϕ F to the variety of in (C × ) n .This sum of the terms of f whose exponents lie in F is a facial form of f and is written f | F .Given a system Φ : of their facial forms is the facial system Φ| F of Φ.
Corollary 3.5.Let M be the intersection of the hyperplanes given by the polynomials in a system Φ of Laurent polynomials with support A. For each face F of conv(A), the points of X • F ∩ M pull back under ϕ F to the solutions of the facial system Φ| F .If no facial system Φ| F has a solution, then the number of solutions to Φ = 0 on (C × ) n is n!vol(conv(A)).
Proof.The first statement follows from the observation about a single polynomial f and its facial form f | F , and the second is a consequence of a version of Corollary 3.3 for The second statement is essentially [3, Thm.B] and is also explained in [30,Sect. 3.4].

3.2.
Facial systems of the critical point equations.We prove Theorem B from the Introduction by interpreting the facial systems of the critical point equations.It is useful to introduce the following notion.A polynomial f (x) in x ∈ (C × ) n is quasi-homogeneous with quasi-homogeneity w ∈ Z n if there is a number 0 = w f such that Equivalently, f is quasi-homogeneous if its support A(f ) lies on a hyperplane not containing the origin.The quasi-homogeneities of f are those w ∈ Z n whose dot product is constant on A(f ).For t ∈ C × and w ∈ Z n , let t w := (t w 1 , . . ., t wn ) ∈ (C × ) n .Lemma 3.6.Suppose that f has a quasi-homogeneity w ∈ Z n .Then (1) For t ∈ C × and x ∈ (C × ) n , we have (2) We have Proof.Note that for a ∈ Z n , (t w • x) a = t w•a x a .The first statement follows.For the second, note that and Newton polytope P := conv(A).We will assume that P has dimension d+1, and also that A ∩ Z d × {0} is a facet of A, called its base.Let (9) be the critical point equations for λ on ψ and M ψ ⊂ P A the corresponding linear subspace of codimension d+1.
Let 0 := 0 d in Z d and e := (0, 1).The base of A is exposed by e and it is the support of ψ(z, 0).A main difference between the sparse equations of Section 3.1 and the critical point equations ( 8) is that the critical point equations allow solutions with λ = 0, which is the component of the boundary of the toric variety corresponding to the base of A. A face F of P is vertical if it contains a vertical line segment, one parallel to e. Lemma 3.7.Suppose that F is a proper face of P that is not the base of P and is not vertical.Then the corresponding facial system of the critical point equations has a solution if and only if the hypersurface Var(ψ| F ) defined by ψ| F in (C × ) d+1 is singular.
Proof.Let 0 = w ∈ Z d+1 be an integer vector that exposes the face F .As F is not vertical we may assume that w d+1 is nonzero.As is not the base, it lies on an affine hyperplane that does not contain the origin, so that ψ| F is quasi-homogeneous with some quasi-homogeneity w.Write w F for the constant w • a for a ∈ F .By Lemma 3.6 (2), we have (11) w Suppose now that (z, λ) is a solution of the restriction of the critical point equations to the face F .That is, at (z, λ), Observe that (z i ∂ψ ∂z i (and the same for λ).Since w d+1 = 0, these equations and (11) together imply that (λ ∂ψ ∂λ )| F = 0, which implies that (z, λ) is a singular point of the hypersurface Var(ψ| F ) defined by ψ| F .
We deduce the following theorem.We apply this when ψ is the dispersion function D(z, λ).Recall that the boundary of the variety X A (X D ) corresponds to all proper faces of its Newton polytope N (D), except for its base.We deduce the following precise version of Theorem B. Corollary 3.9.Let L be an operator on a periodic graph and set D = det(L(z) − λI).If N (D) has no vertical faces and if for each face F that is not its base, Var(D| F ) is smooth, then the Bloch variety has exactly (d+1)!vol(conv(A(D))) critical points.
Example 3.10.The restriction on vertical faces is necessary.General operators on the second graph in Figure 2 (an abelian cover of K 4 ) have the following Newton polytope: It has base [−1, 1] 2 , apex (0, 0, 4), and the remaining vertices are at (±1, 0, 1) and (0, ±1, 1).It has volume 20/3, so we expect 40 = 3! • 20/3 critical points.However, there are at most 32 critical points, as direct computation shows that the critical point equations have two solutions on each of its four vertical faces.⋄

Newton polytopes and dense periodic graphs
The Newton polytope N (D) of the dispersion function of an operator on a periodic graph is central to our results.In Section 4.1 we associate a polytope N (Γ) to any periodic graph Γ such that N (D) ⊂ N (Γ) for any operator on Γ, and that we have equality for almost all parameter values.We call N (Γ) the Newton polytope of Γ.
A periodic graph Γ is dense if it has every possible edge, given its support A(Γ) and fundamental domain W . Every periodic graph is a subgraph of a minimal dense periodic graph.We identify the Newton polytope of a dense periodic graph and show that when d = 2 or 3, a general operator on Γ satisfies Corollary 3.9, which implies Theorem C.
Let Γ be a connected Z d -periodic graph with fundamental domain W .Its support A(Γ) is the finite set of points a ∈ Z d such that there is an edge between W and a+W .The integer span of A(Γ) is Z d , as Γ is connected.The graph Γ is dense if for every a ∈ A(Γ), there is an edge in Γ between every pair of vertices in the union of W and a+W .In particular, the restriction of Γ to W is the complete graph on W .The graphs of Figures 1 and 6 are dense, while those of Figure 2 are not dense.The set of parameters (e, V ) for operators on a periodic graph Γ is Y = C E × C W , where E is the set of orbits of edges.We observed that for any c ∈ Y , each entry of L c (z) has support a subset of A(Γ).Consequently, each diagonal entry of L c (z) − λI has support a subset of A(Γ) ∪ {e} and its Newton polytope is a subpolytope of Q := conv(A(Γ) ∪ {e}).Let m := |W |, the number of orbits of vertices.Figure 7 shows mQ = 2Q for the dense graphs of Figures 1 and 6.Observe that mQ is a Together with Corollary 3.9, this implies Theorem C from the Introduction.We prove Theorem 4.2 in the following two subsections.4.1.The Newton polytope of Γ.For a periodic graph Γ, the space of parameters (e, V ) for operators on Γ is Y = C E × C W . Treating parameters as indeterminates gives the generic dispersion function D(e, V, z, λ), which is a polynomial in z, λ whose coefficients are polynomials in the parameters e, V .The Newton polytope N (Γ) of Γ is the convex hull of the monomials in z, λ that appear in D(e, V, z, λ).
Proof.For any c = (e, V ) ∈ Y , D c (z, λ) is the evaluation of the generic dispersion function D(e, V, z, λ) at the point (e, V ).Thus N (D c ) ⊂ N (Γ).
The coefficient C (a,j) of a monomial z a λ j in D(e, V, z, λ) is a polynomial in (e, V ).For any c = (e, V ) ∈ Y , z a λ j appears in D c if any only if C (a,j) (e, V ) = 0. Thus, we have the equality N (D c ) = N (Γ) of Newton polytopes if and only if C (a,j) (e, V ) = 0 for every vertex (a, j) of N (Γ), which defines a dense open subset U ⊂ Y .
When Γ is dense and no parameter c vanishes, then every diagonal entry of L c (z) − λI has support A(Γ) ∪ {e}.This implies that N (Γ) = mQ.4.2.Smoothness of the Bloch variety at infinity.Let Γ be a dense periodic graph with d = 2 or 3. Let U ⊂ Y be the subset of Lemma 4.3.We show that for each face F of N (Γ) that is not its base, there is a nonempty open subset U F of U such that for c ∈ U F , the restriction D c | F to the monomials in F defines a smooth hypersurface.Then for parameters c in the intersection of the U F , the operator satisfies the hypotheses of Corollary 3.9, which proves Theorem 4.2 and Theorem C.
Let F be a face of N (Γ) that is not its base and let c ∈ U. We may assume that F is not a vertex, for then where each entry of the matrix (L c (z Since G is not the base of Q (and thus does not contain the origin), we make the following observation, which follows from the form of the operator L c (2).If the apex e = (0, 1) of Q lies in G and f is a diagonal entry of (L c (z) − λI)| G , then f contains the term −λ.Any other integer point a ∈ G is nonzero and lies in the support A(Γ) of Γ, and the coefficient of z a in f is −e (u,a+v) , where f is the entry in row u and column v. Consequently, except possibly for terms −λ, all coefficients of entries in (L c (z) − λI)| G are distinct parameters.
Suppose that the fundamental domain is W = {v 1 , . . ., v m } so that we may index the rows and columns of L c (z) by 1, . . ., m.Let Y ′ ⊂ Y be the set of parameters c where e (v i ,a+v j ) = 0 if a ∈ G and j = i, i+1.
(Here, m+1 is interpreted to be 1.)For c ∈ Y ′ , all entries of L c (z)| G are zero, except on the diagonal, the first super diagonal, and the lower left entry.The same arguments as in the proof of Lemma 4.3 show that there exist parameters c ∈ Y ′ such that D c (z, λ) has Newton polytope N (Γ).Thus Y ′ ∩ U = ∅, where U ⊂ Y is the set of Lemma 4.3.

Theorem 4.4.
There exists an open subset ) is an open condition on the space Y of parameters, this will complete the proof of Theorem 4.2, and thus also of Theorem C.
Proof.Let us write ψ c (z, λ) for the facial polynomial D c (z, λ)| F .We will show that the set of c ∈ Y ′ such that Var(ψ c (z, λ)) is singular is a finite union of proper algebraic subvarieties.As c ∈ Y ′ , the only nonzero entries in the matrix (L c (z) − λI)| G are its diagonal entries f 1 (z, λ), . . ., f m (z, λ) and the entries g 1 (z), . . ., g m (z) which are in positions (1, 2), . . ., (m−1, m) and (m, 1), respectively.Thus For a polynomial f in the variables (z, λ), write ∇ T for the toric gradient operator, Note that Here f i indicates that f i does not appear in the product, and the same for g i .Let (z, λ) ∈ Var (ψ c ) be a singular point.Then ψ c (z, λ) = 0 and ∇ T ψ c (z, λ) = 0.There are five cases that depend upon the number of polynomials f i , g j vanishing at (z, λ).
(i) At least two polynomials f p and f q and two polynomials g r and g s vanish at (z, λ).Thus ψ(z, λ) = 0 and by (12) this implies that ∇ T ψ c (z, λ) = 0. (ii) At least two polynomials f p and f q and exactly one polynomial g s vanish at (z, λ).
Thus ψ(z, λ) = 0 and by ( 12) if ∇ T ψ c (z, λ) = 0, then ∇ T f p (z, λ) = 0. (iv) Exactly one polynomial f p and one polynomial g r vanish at (z, λ).Thus ψ(z, λ) = 0 and by ( 12) if ∇ T ψ c (z, λ) = 0, then, after reindexing so that p = r = 1, we have (v) No polynomials f i or g i vanish at (z, λ).In each case, we will show that the set of parameters c ∈ Y ′ such that there exist (z, λ) satisfying these conditions lies in a proper subvariety of Y ′ .Cases (i)-(iv) use arguments based on the dimension of fibers and images of a map and are proven in the rest of this section.Case (v) is proven in Section 4.3 and it uses Bertini's Theorem.
Let us write X for the space (C × ) d+1 and x for a point (z, λ) ∈ X.We first derive consequences of some vanishing statements.For a finite set F ⊂ Z d+1 , let C F be the space of coefficients of polynomials in x ∈ X with support F .This is the parameter space for polynomials with support F .Lemma 4.5.We have the following.
(1) For any x ∈ X, f (x) = 0 is a nonzero homogeneous linear equation on Suppose that the affine span of F does not contain the origin.Then (3) For any f ∈ C F and x ∈ X, ∇ T f = 0 implies that f (x) = 0.
(4) For any x ∈ X, the equation ∇ T f (x) = 0 defines a linear subspace of C F of codimension dim CF .
Proof.Writing f = a∈F c a x a , the first statement is obvious.We have ∇ T f = ac a x a .As the coefficients c a are independent complex numbers and x a = 0, Statement (2) is immediate.The hypothesis that the affine span of F does not contain the origin implies that any f ∈ C F is quasi-homogeneous.Statement (3) follows from Equation (11).The last statement follows from the observation that the set of f such that ∇ T f = 0 is the kernel of a surjective linear map C F ։ CF .
Let F := G ∩ (A(Γ) ∪ {e}), where e = (0, 1), be the (common) support of the diagonal polynomials f i and let G := G ∩ A(Γ) be the (common) support of the polynomials g j .We either have that F = G or F = G ∪ {e}.Also, |F | > 1 as G is not a vertex, and as G is a proper face of Q = conv(A(Γ) ∪ {e}), but not its base, the polynomials f i , g j are quasi-homogeneous with a common quasi-homogeneity.
The parameter space for the entries of (L c (z We write c = (f • , g • ) = (f 1 , . . ., f m , g 1 , . . ., g m ) for points of Z.This is a coordinate subspace of the parameter space Y ′ .As Z contains exactly those parameters that can appear in the facial polynomial ψ c (x), it suffices to show that the set of parameters c = (f lies in a proper subvariety of Z.
Proof.Consider the incidence correspondence, This has projections to X and to Z and its image in Z is the set Θ. Consider the projection π X : Υ → X.By Lemma 4.5 (1), for x ∈ X, each condition f i (x) = 0, g i (x) = 0 for i = 1, 2 is a linear equation on C F or C G .These are independent on Z as they involve different variables.Thus the fiber π −1 X (x) is a vector subspace of Z of codimension 4, and dim(Υ) = dim(Z) − 4 + dim(X) = dim(Z) + d − 3.
Consider the projection π Z to Z and let (f • , g • ) ∈ π Z (Υ).Then there is an x ∈ X such that f 1 (x) = f 2 (x) = g 1 (x) = g 2 (x) = 0. Let w ∈ Z d+1 be a common quasi-homogeneity of the polynomials f i , g j .By Lemma 3.6 (1), for any t ∈ C × , each of f 1 , f 2 , g 1 , g 2 vanishes at t w • x.Thus the fiber π −1 Z (f • , g • ) has dimension at least one.By the Theorem [29, Theorem 1.25] on the dimension of the image and fibers of a map, the image π Z (Υ) has dimension at most dim(Z) + d − 4 < dim(Z), which establishes the lemma.
After reindexing and possibly interchanging f with g, Cases (ii) and (iii) in the proof of Theorem 4.4 follow from the next lemma.
lies in a proper subvariety of Z.
Proof.Consider the incidence correspondence, Let x ∈ X and consider the fiber π −1 X (x).As in the proof of Lemma 4.6, the conditions f 1 (x) = f 2 (x) = 0 are two independent linear equations on Z.By Lemma 4.5 (3), ∇ T g 1 (x) = 0 implies that g 1 (x) = 0, and by Lemma 4.5 (4), the condition ∇ T g 1 (x) = 0 is dim CG further independent linear equations on Z.
If |G| = 1, so that g 1 = c a x a is a single term, then g(x) = 0 implies that c a = 0. Consequently, the image Θ of Υ in Z lies in a proper subvariety.Otherwise, |G| > 1 which implies that dim CG ≥ 2, and thus the fiber has codimension at least 4. As in the proof of Lemma 4.6, this implies that Θ lies in a proper subvariety of Z.
Case (iv) in the proof of Theorem 4.4 is more involved.
Lemma 4.8.The set lies in a proper subvariety of Z.
Proof.The set Θ includes the sets of Lemmas 4.6 and 4.7.Let Θ • ⊂ Θ be the set of c = (f • , g • ) that have a witness x ∈ X (f 1 (x) = g 1 (x) = 0 and ∇ T ψ c (x) = 0) such that none of ∇ T f 1 (x), ∇ T g 1 (x), or f i (x)g i (x) for i > 1 vanish.It will suffice to show that Θ • lies in a proper subvariety of Z.
We show that Θ Dividing (13) by m i=2 f i (x) gives ∇ T f 1 (x)−(−1) m y∇ T g 1 (x) = 0, and thus (y, x, f • , g • ) ∈ Υ.We now determine the dimension of Υ.Let (y, x) ∈ C × × X and consider the fiber π −1 (y, x) ⊂ Z above it in Υ.The two linear and one nonlinear equations ( 14) are independent on Z as they involve disjoint sets of variables, and thus define a subvariety T ⊂ Z of codimension 3. Consider the remaining equation, ∇ T f 1 (x) − (−1) m y∇ T g 1 (x) = 0. Note that if e = (0, 1) lies in the support F of f 1 , so that F = G ∪ {e}, then ∇ T f 1 (x) contains the term −e and thus cannot lie in the span CG of G, which contains ∇ T g 1 (x) by Lemma 4.5 (2).In this case the fiber is empty and Θ Proof of Lemma 4.10.Let T ⊂ X × Z be the set of (x, f We claim that ϕ(T ) is dense in C × × Z.For this, recall that the polynomials f i have support F , which is G ∩ (A(Γ) ∪ {e}) for some face G of Q = conv(A(Γ) ∪ {e}) that is neither its base nor a vertex, and the polynomials g i have support G = G ∩ A(Γ).Since G is not a vertex, there are a, b ∈ F with a = b and b ∈ A(Γ).
Let f i := x a and g The map X = (C × ) d+1 → C × given by x → x ma − (−1) m x mb is surjective as ma − mb = 0.This implies that the differential dϕ is surjective at any point of X × {(f

Critical points property
We illustrate our results, using them to establish the critical points property (and thus the spectral edges nondegeneracy conjecture) for three periodic graphs.We first state this property.
Let Γ be a connected Z d -periodic graph with parameter space Y = C E × C W for discrete operators on Γ.We say that Γ has the critical points property if there is a dense open subset U ⊂ Y such that if c ∈ U, then every critical point of the function λ on the Bloch variety Var(D c (z, λ)) is nondegenerate in that the Hessian determinant is nonzero at that critical point.Here, the derivatives are implicit, using that D(z, λ) = 0. Implicit differentiation of D = 0 gives ∂D ∂z j + ∂D ∂λ • ∂λ ∂z j = 0.If ∂D ∂λ = 0, then ∂λ ∂z j = 0.If ∂D ∂λ = 0, then (z, λ) is a singular point hence is also a critical point of the function λ and so we again have ∂λ ∂z j = 0. Differentiating again we obtain, At a critical point (so that ∂λ ∂z j = 0), we have .
Consider now the Jacobian matrix of the critical point equations ( 16), At a critical point, the first row is (0 .
A solution ζ of a system of polynomial equations on C n is regular if the Jacobian of the system at ζ has full rank n.Regular solutions are isolated and have multiplicity 1.We deduce the following lemma.Proof.Let Y be the parameter space for operators L on Γ.Consider the variety which is the incidence variety of critical points on all Bloch varieties for operators on Γ.Let π be its projection to Y .For any c ∈ Y , the fiber π −1 (c) is the set of critical points of the function λ on the corresponding Bloch variety for D c .By Corollary 2.5, there are at most (d+1)!vol(N (D c )) isolated points in the fiber.
Let c ∈ Y be a point such that the critical point equations have (d+1)!vol(N(Γ)) regular solutions.Then (d+1)!vol(N (Γ)) ≤ (d+1)!vol(N (D c )).By Lemma 4.3, N (D c ) is a subpolytope of N (Γ), so that vol(N (D c )) ≤ vol(N (Γ)).We conclude that both polytopes have the same volume and are therefore equal.In particular, the corresponding Bloch variety has the maximum number of critical points, and each is a regular solution of the critical point equations (8).Because they are regular solutions, the implicit function theorem implies that there is a neighborhood U c of c in the classical topology on Y such that the map π −1 (U c ) → U c is proper (it is a (d+1)!vol(N (Γ))-sheeted cover).
The set DC of degenerate critical points is the closed subset of CP given by the vanishing of the Hessian determinant (15) and two vertices in W , and independent computations in the computer algebra systems Macaulay2 [17] and Singular [8] find a point c ∈ Y = C 15 such that the critical point equations have 64 regular solutions on (C × )2 × C. By Theorem 5.2, the critical points property holds for Γ.These computations are independent in that the code, authors, and parameter values for each are distinct.⋄ Example 5.4.The graph Γ in Figure 9 is not dense.Its restriction to the fundamental domain is not the complete graph on 3 vertices and there are three and not nine edges between any two adjacent translates of the fundamental domain.Altogether, it has 3 • 6 + 1 = 19 fewer edges than the corresponding dense graph.Its support A(Γ) forms the columns of the matrix ( 0 1 1 0 −1 −1 0 0 0 1 1 0 −1 −1 ) whose convex hull is a hexagon of area 3. Despite Γ not being dense, its Newton polytope N (Γ) is equal to the Newton polytope of the dense graph with the same parameters, A(Γ) and W . Figure 9 displays the Newton polytope, along with elements of the support of the dispersion function that are visible.Observe that on each triangular face, there are four and not ten monomials.
By Theorem A (Corollary 2.5), there are at most 2! • 3 2+1 • 3 = 162 critical points.There are eleven edges and three vertices in W , and independent computations in Macaulay2 and Singular find a point c ∈ Y = C 14 such that the critical point equations have 162 regular solutions on (C × ) 2 × C. By Theorem 5.2, the critical points property holds for Γ.
Let Γ ′ be a graph that has the same vertex set and support as Γ, and contains all the edges of Γ-then [9,Thm. 22] implies that the critical points property also holds for Γ ′ .This establishes the critical points property for an additional 2 19

Conclusion
We considered the critical points of the complex Bloch variety for an operator on a periodic graph.We gave a bound on the number of critical points-the normalized volume of a Newton polytope-together with a criterion for when that bound is attained.We presented a class of graphs (dense periodic graphs) and showed that this criterion holds for general discrete operators on a dense graph.Lastly, we used these results to find 2 19 + 2 graphs on which the spectral edges conjecture holds for general discrete operators when d = 2.

Theorem A .
The function λ on the Bloch variety of a discrete operator on Γ has at most d! |W | d+1 vol(A(Γ)) isolated critical points.Here, vol(A(Γ)) is the Euclidean volume of the convex hull of A(Γ).

Figure 1 .
Figure 1.A dense periodic graph Γ with the convex hull of A(Γ).

Figure 2
shows two

Figure 3 .
Figure 3.A labeling of the hexagonal lattice.

Figure 4 .
Figure 4.A Bloch variety and spectral bands for the hexagonal lattice.
point of the function λ on the Bloch variety Var (D(z, λ)) if and only if (8) holds.

Theorem 3 . 8 .
If the Newton polytope N (ψ) of ψ has no vertical faces and the restriction of ψ to each face that is not the base of N (ψ) defines a smooth variety, then the critical point equations have exactly (d+1)!vol(N (A)) solutions in (C × ) d × C.

Figure 6 .
Figure 6.A dense graph Γ and its support A(Γ) with convex hull.

Lemma 4 . 1 .
The Newton polytope N (D c ) is a subpolytope of the dilation mQ of Q.Proof.The dispersion function D c is a sum of products of m entries of the m × m matrix L c (z) − λI.Each such product has Newton polytope a subpolytope of mQ as the Newton polytope of a product is the sum of Newton polytopes of the factors.

Figure 7 .
Figure 7. Newton polytopes of dense graphs.pyramid with base m conv(A(Γ)) and apex me, and it has no vertical faces.Theorem 4.2.Let Γ be a dense Z d -periodic graph.There is a nonempty Zariski open subset U of the parameter space Y such that for c ∈ U, the Newton polytope of D c (z, λ) is the pyramid mQ.When d = 2 or 3, then we may choose U so that for every c ∈ U and face F of mQ that is not its base, Var(D c | F ) is smooth.

5. 1 .
Reformulation of Hessian condition.Let D = det(L c (z) − λI) be the dispersion function for an operator L c on a periodic graph Γ.In Section 2 we derived the equations for the critical points of the function λ on the Bloch variety Var (D(z, λ)),

Lemma 5 . 1 .
A nonsingular critical point (z, λ) on Var(D c (z, λ)) is nondegenerate if and only if it is a regular solution of the critical point equations(16).The following theorem is adapted from arguments in [9, Sect.5.4].Theorem 5.2.Let Γ be a Z d -periodic graph.If there is a parameter value c ∈ Y such that the critical point equations have (d+1)!vol(N(Γ)) regular solutions, then the critical points property holds for Γ.

Figure 8 .
Figure 8. Dense periodic graph and its polytope from Figure 6.

Figure 9 .
Figure 9. Sparse graph with the same Newton polytope as the corresponding dense graph.
• , g • ) ∈ Z such that Var (ψ c (x)) is singular lies in a proper subvariety of Z.The same case distinctions (i)-(v) in the proof of Theorem 4.4 apply.After reindexing, Case (i) in the proof of Theorem 4.4 follows from the next lemma.
• , g • )}, and therefore ϕ(T ) is dense in C × × Z. Since T is an open subset of the smooth variety X × Z, it is smooth.Then Bertini's Theorem [29, Thm.2.27, p. 139] implies that there is a dense open