On Circulant Matrices Irwin Kra and Santiago R. Simanca
ome mathematical topics—circulant ma- years, the authors’ interest was rekindled by the trices, in particular—are pure gems that casual discovery by S. Simanca that these matrices cry out to be admired and studied with are connected with algebraic geometry over the different techniques or perspectives in mythical field of one element. mind. Circulant matrices are prevalent in many parts SOur work on this subject was originally moti- of mathematics (see, for example, [8]). We point vated by the apparent need of the first author to the reader to the elegant treatment given in [4, derive a specific result, in the spirit of Proposition §5.2] and to the monograph [1] devoted to the 24, to be applied in his investigation of theta subject. These matrices appear naturally in areas constant identities [9]. Although progress on that of mathematics where the roots of unity play a front eliminated the need for such a theorem, role, and some of the reasons for this will unfurl the search for it continued and was stimulated in our presentation. However ubiquitous they are, by enlightening conversations with Yum-Tong Siu many facts about these matrices can be proven during a visit to Vietnam. Upon the first author’s using only basic linear algebra. This makes the return to the U.S., a visit by Paul Fuhrmann brought area quite accessible to undergraduates looking to his attention a vast literature on the subject, for research problems or mathematics teachers including the monograph [4]. Conversations in searching for topics of unique interest to present the Stony Brook mathematics common room at- to their students. tracted the attention of the second author and We concentrate on the discussion of necessary that of Sorin Popescu and Daryl Geller∗ to the and sufficient conditions for circulant matrices to subject and made it apparent that circulant matri- be nonsingular and on various distinct represen- ces are worth studying in their own right, in part tations they have, goals that allow us to lay out because of the rich literature on the subject con- the rich mathematical structure that surrounds necting it to diverse parts of mathematics. These them. Our treatment, though, is by no means ex- productive interchanges between the participants haustive. We expand on their connection to the resulted in [5], the basis for this article. After that algebraic geometry over a field with one element, version of the paper lay dormant for a number of to normal curves, and to Toeplitz’s operators. Irwin Kra is emeritus professor of mathematics at State The latter material illustrates the strong presence University of New York at Stony Brook. His email address these matrices have in various parts of modern is [email protected]. and classical mathematics. Additional connections Santiago R. Simanca is a visiting senior professor of to other mathematics may be found in [11]. mathematics at the Laboratoire Jean Leray, Nantes, The paper is organized as follows. In the France. His email address is [email protected]. section “Basic Properties” we introduce the ba-
∗Our colleague Daryl died on February 5, 2011. We dedi- sic definitions and present two models of the cate this manuscript to his memory. space of circulant matrices, including that as a finite-dimensional commutative algebra. Their de- DOI: http://dx.doi.org/10.1090/noti804 terminant and eigenvalues, as well as some of
368 Notices of the AMS Volume 59, Number 3 th k 1 their other invariants, are computed in the section operator acting on v; its k row is T − v, k = “Determinants and Eigenvalues”. In the section 1,...,n: “The Space of Circulant Matrices” we discuss fur- v0 v1 vn 2 vn 1 ther the space of such matrices and present their − − vn 1 v0 vn 3 vn 2 third model identifying them with the space of − − − ...... diagonal matrices. In the section “Roots of Poly- V . . . . . . = nomials” we discuss their use in the solvability of v2 v3 v0 v1 polynomial equations. All of this material is well v1 v2 vn 1 v0 − known. Not so readily found in the literature is We denote by Circ(n) Mn the set of all n n the remaining material, which is also less elemen- complex circulant matrices.⊂ × tary. In the section “Singular Circulant Matrices” It is obvious that Circ(n) is an n-dimensional we determine necessary and sufficient conditions complex vector space (the matrix V is identified for classes of circulant matrices to be nonsingu- with its first row) under the usual operations of lar. The geometry of the affine variety defined by matrix addition and multiplication of matrices these matrices is discussed in the section “The by scalars; hence our first model for circulant Geometry of Circ(n)”, where we also speculate on matrices is provided by the C-linear isomorphism some fascinating connections. In the section “The (FIRST MODEL) I : Circ(n) Cn, Rational Normal Curves Connection” we establish → a relationship between the determinant of a cir- where I sends a matrix to its first row. Matrices culant matrix and the rational normal curve in can, of course, be multiplied, and one can easily complex projective space and uncover their con- check that the product of two circulant matrices nection to Hankel matrices. Finally, in the section is again circulant and that for this set of matrices, “Other Connections—Toeplitz Operators” we re- multiplication is commutative. However, we will late them to the much-studied Toeplitz operators shortly see much more and conclude that we are and Toeplitz matrices as we outline their use in dealing with a mathematical gem. Before that we an elementary proof of Szegö’s theorem. record some basic facts about complex Euclidean It is a pleasure for the first author to thank Yum- space that we will use. Tong Siu for outlining another elementary proof of The ordered n-tuples of complex numbers can formula (3) and for generating his interest in this be viewed as the elements of the inner product n topic. He also thanks Paul Fuhrmann for bringing space C with its Euclidean (L2-norm) and standard to his attention a number of references on the orthonormal basis subject and for the helpful criticism of an earlier ei (δi,0,...,δi,n 1) , i 0,...,n 1, draft of this manuscript. It is with equal pleasure = − = − where δi,j is the Kronecker delta ( 1 for i j and that the second author thanks A. Buium for many = = conversations about the subject of the field with 0 for i ≠ j). We will denote this basis by e and one element and for the long list of related topics remind the reader that in the usual representation n 1 that he brought to his attention. v (v0, v1,...,vn 1) i −0 viei , the vi ’s are the = − = = components of v with respect to the basis e. Basic Properties To explore another basis, we fix once and for all a choice of a primitive nth root of unity We fix hereafter a positive integer n 2. Our ≥ 2πı main actors are the n-dimensional complex vector ǫ e n , space Cn and the ring of n n complex matrices = × define for l 0, 1,...,n 1, Mn. We will be studying the multiplication Mv of = − n matrices M Mn by vectors v C . In this regard, 1 l 2l (n 1)l n ∈ ∈ xl (1, ǫ , ǫ ,...,ǫ − ) C , we view v as a column vector. However, at times, = √n ∈ it is useful mathematically and more convenient and introduce a special case of the Vandermonde typographically to consider matrix
v (v0, v1,...,vn 1) Cn 1 1 1 1 = − ∈ n 2 n 1 1 ǫ ǫ − ǫ − n as a row vector. We define a shift operator T : C 1 . . . . . Cn → E ...... by = √n n 2 (n 2)2 (n 2)(n 1) 1 ǫ − ǫ − ǫ − − T (v0,v1,...,vn 1) (vn 1,v0,...,vn 2). n 1 (n 1)(n 2) (n 1)2 − = − − 1 ǫ − ǫ − − ǫ − We start with the basic and key definition. It is well known and established by a calculation Definition 1. The circulant matrix V circ v as- that n = { } sociated to the vector v C is the n n matrix n j i ∈ × (1) det E n− 2 (ǫ ǫ ) ≠ 0; whose rows are given by iterations of the shift = 0 i March 2012 Notices of the AMS 369 hence E is nonsingular. In fact, E is a most Remark 5. The algebra C[X]/(Xn 1) can be iden- 1 t − remarkable matrix: It is unitary, E− E , and it tified with the space Pn 1 of complex polynomials − t 1 = is symmetric, E E, and hence E− E; and its of degree (n 1) with appropriate definition of = = ≤ − columns and rows are the vectors xl . multiplication of its elements. Under this identifi- We view E as a self-map of Cn{ and} conclude cation, for V Circ(n), ∈ that Eei E(ei ) xi . Since E is nonsingular, we J(V) PV (X). = = = see that the xl are another orthonormal basis n { } for C , to be denoted by x. The C-linear self-map Determinants and Eigenvalues n of C defined by the matrix E depends, of course, The Basic Theorem on the bases for the domain and target; to show Many results about circulants follow from direct this dependence, the map should be denoted as calculations; in particular, the next theorem. Ee,e. Observe that as linear maps, Ee,e Ie,x, where = n I is the n n identity matrix. Theorem 6. If v (v0,v1,..., vn 1) C and V = − ∈ = To return× to circulant matrices, we let circ v , then { } n 1 n 1 n 1 Wi circ ei , 0 i n 1. − − − = { } ≤ ≤ − (3) det V ǫjl v P (ǫl ). j V Itisobviousthatwehavea standard representation = l 0 j 0 = l 0 = = = or form of circulant matrices: Proof. We view the matrix V as a self-map Ve,e of n 1 n − C . For each integer l, 0 l n 1, let∗ circ (v0,v1,...,vn 1) viWi . ≤ ≤ − − l (n 1)l l { } = i 0 λl v0 ǫ v1 ǫ − vn 1 PV (ǫ ). = = + + + − = It is less obvious, but follows by an easy calcula- A calculation shows that Vxl λl xl. Thus λl is = tion, that Wi Wj Wi j , where all the indices are an eigenvalue of V with normalized eigenvector = + interpreted mod n. Obviously W0 I, and letting xl. Since, by (1), x0, x1,...,xn 1 is a linearly inde- { n − } i = C W W1, we see that W Wi . pendent set of vectors in , the diagonal matrix = = with the corresponding eigenvalues is conjugate Remark 2. With respect to the standard basis of n 1 n to V , and we conclude that det V l −0 λl. C , the shift operator T is represented by the = = transpose W t of the matrix W . Note that (W i )t Corollary 7. All circulant matrices have the same n i = W . ordered set of orthonormal eigenvectors xl . − { } It is useful to introduce Corollary 8. The characteristic polynomial pV of Definition 3. The (polynomial in the indetermi- V is given by nate X) representer PV of the circulant matrix (4) n 1 0 − V circ (v0,v1,...,vn 1) is n i = { − } pV (X) det (XI V) (X λl ) X biX . n 1 = − = − = + − l 0 i n 1 i = = − (2) PV (X) vi X . (Here we let the last equality define the bi ’s as func- = i 0 = tions of the λl ’s. They are the elementary symmetric As usual, we let C[X] denote the ring of complex functions of the λl ’s.) C polynomials and for f (X) [X], (f (X)), the Corollary 9. The nullity of V Circ(n) is the num- principal ideal generated by∈ this polynomial. We ∈ ber of zero eigenvalues λl. have established most of the following theorem (the remaining claims are easily verified). We use similar symbols for the characteristic polynomial pV of a circulant matrix V and its Theorem 4. Circ(n) is a commutative algebra that representer PV . They exhibit, however, different is generated (over C) by the single matrix W . The relations to V . map that sends W to the indeterminate X extends If we let ν(V) denote the nullity of V , the last by linearity and multiplicativity to an isomorphism corollary can be restated as of C-algebras n For all V Circ(n), ν(V) deg gcd(pV (X), X ). (SECOND MODEL) J : Circ(n) C[X]/(Xn 1). ∈ = → − Corollary 10. Let V be a circulant matrix with rep- The map that sends a circulant matrix V to its resenter PV . The following are equivalent: t transpose V is an involution of Circ(n) and (a) The matrix V is singular. corresponds under J to the automorphism of l (b) PV ǫ 0 for some l Z. n n 1 C[X]/(X 1) induced by X ֏ X − . = ∈ n − (c) The polynomials PV (X) and X 1 are not relatively prime. − Proof. The only nontrivial observation is that multiplication of circulant matrices in stan- ∗Throughout this paper the symbols λl and xl are re- dard form corresponds to the multiplication in served for the eigenvalue and eigenvectors we introduce C[X]/(Xn 1). here. − 370 Notices of the AMS Volume 59, Number 3 Again, we have a reformulation of part of the The Space of Circulant Matrices last corollary as To obtain our third model for Circ(n), we start n by defining Dn to be the space of n n diagonal For all V Circ(n), ν(V) deg gcd(PV (X), X 1). × ∈ = − matrices. This space is clearly linearly isomorphic to Cn. Remark 11. The shift operator T acts on Circ(n). If V T V denotes this action, then the traces of Theorem 12. All elements of Circ(n) are simulta- → T k V , k 0,...,n 1, uniquely determine V . The neously diagonalized by the unitary matrix E; that = − representer PV of a circulant matrix V uniquely is, for V in Circ(n), determines and is uniquely determined by the ma- 1 (7) E− VE DV trix. Similarly, the characteristic polynomial and = eigenvalues of a circulant matrix uniquely deter- is a diagonal matrix and the resulting map mine each other. From a given set of ordered eigen- (THIRD MODEL) D : Circ(n) Dn → values, we recover the circulant matrix by the next is a C-algebra isomorphism. theorem. However, a given set of distinct eigenval- ues determines n! circulant matrices. Proof. The n n matrix E represents the linear automorphism× of Cn that sends the unit vector Determinants of Circulant Matrices el to the unit vector xl . If V is a circulant ma- trix and D is the diagonal matrix with diagonal It is easy to see that V entries given by the ordered eigenvalues of V : detcirc (v0,v1,...,vn 1) λ0,λ1,...,λn 2,λn 1, then (7) holds. The map { − } − − 1 n 1 D is onto, because for all D Dn, EDE− is ( 1) − det circ (v1,v2,...,vn 1,v0) ∈ = − { − } circulant. n 1 ( 1) − det circ (vn 1,vn 2,...,v1,v0) , = − { − − } Corollary 13. The inverse of an invertible element and iterations of these yield that det V of Circ(n) also belongs to Circ(n). k(n 1) k = ( 1) − det T V for each integer 0 k < n. − ≤ Proof. If V is a nonsingular circulant matrix, then However, there is no obvious general relation 1 DV is invertible and D− D 1 . between V = V − Corollary 14. The characteristic polynomial of detcirc (v0,v1,...,vn 1) { − } V Circ(n) is given by and ∈ pV (X) det (XI V) det(XI DV ). detcirc (vσ(0),vσ(1),...,vσ(n 1)) = − = − { − } for σ Sn, the permutation group on n letters. Remark 15. The last corollary encodes several For example,∈ facts that can be established by other methods: i 3 3 3 Let p Pn 1. If p(X) aiX and (5) detcirc (v0,v1,v2) v0 v1 v2 3v0v1v2, • ∈l − = { } = + + − λl p(ǫ ), then the elementary symmetric = a function that is invariant under the permutation functions of the λl’s belong to the ring group S3, while generated (over Z) by the ai ’s. detcirc (v ,v ,v ,v ) Given an ordered set λl , the unique poly- 0 1 2 3 • { } l { } nomial p satisfying λl p(ǫ ) is det(XI (6) (v0 v1 v2 v3)(v0 v1 v2 v3) = − = + + + − + − DV ). 2 2 ((v0 v2) (v1 v3) ), × − + − which, though admitting some symmetries, fails Roots of Polynomials to be invariant under the action of the entire Each n n circulant matrix V has two polynomials × group S4; for instance, it is not invariant under naturally associated to it: its representer PV and the transposition that exchanges v0 and v1. its characteristic polynomial pV . These are both The action of C× on Circ(n) by dilations can be described explicitly in terms of the eigenvalues λl used to understand further the singular circulant of V . The characteristic polynomial pV is the unique monic polynomial of degree n that vanishes at each matrices. For given a C×, we have that ∈ λl . The representer PV is the unique polynomial of l detcirc (av0,av1,...,avn 1) degree n 1 whose value at ǫ is λl. { − } ≤ − n The roots of the characteristic polynomial of a detcirc (v0,v1,...,vn 1) , = { − } an arbitrary n n matrix V (these are the eigen- and we may cast these matrices as the projective values of the matrix× V ) are obtained by solving a n 1 variety in P − (C) given by the locus of det on monic degree n polynomial equation. In the case Circ(n). The decomposition of this variety into its of circulant matrices, the roots of pV are easily irreducible components yields a geometric inter- calculated using the representer polynomial PV . pretation of the zeroes of various multiplicities of Thus, if a given polynomial p is known to be this function on Circ(n). the characteristic polynomial of a known circulant March 2012 Notices of the AMS 371 matrix V , its zeroes can be readily found. This We now turn our attention to the cases of low remark is the basis of [8] and of the section “Poly- degree. nomials of Degree 4”. Every monic polynomial p is the characteristic≤ polynomial of some circulant Polynomials of Degree 4 ≤ matrix V , and so a very natural problem ensues: Circulant matrices provide a unified approach to If we are given that p pV for a collection of = solving equations of degree 4; we will illustrate circulant matrices V , can we determine one such this for degrees 3 and 4. As≤ is quite common, we V or, equivalently its representer PV directly from start with a definition. p? If so, then the n roots of p are the values of P V Definition 17. Given a monic polynomial p at the n nth roots of unity. of degree n, a circulant n n matrix V × = circ (v0,...,vn 1) is said to adhere to p if The General Case { − } the characteristic polynomial pV of V is equal The vector space Pn 1 of polynomials of degree − to p. n 1 is canonically isomorphic to Circ(n) (both We learned at a quite early age that to solve the ≤ − n are canonically isomorphic as vector spaces to C ). equation Let M be the affine space of monic polynomials of n n 1 n 2 n p(X) X αn 1X − αn 2X − degree n (again identifiable with C ). We define a = + − + − map α1X α0 0, + + + = : Pn 1 M we should use the change of variable Y X − → 1 = + as follows. For each p Pn 1, there exists a unique n αn 1, which eliminates the monomial of degree Λ ∈ − − V Circ(n) such that p PV . Send p to pV . The n 1 in p and leads to the equation ∈ = − map is holomorphic; in fact, it is algebraic. n n 2 q(Y ) Y γn 2Y − γ1Y γ0 0 We have already remarked that it is generically = + − + + + = to be solved. If V circ (v0,v1,...,vn 1) adheres n! to 1.Λ We define three subspaces: = { − } 0 to p, then the traceless matrix V v0I adheres to (1) Pn 1 consisting of those − − q. i j A reasonable program for solving polynomial p Pn 1 with p(ǫ ) p(ǫ ) { ∈ − = equations p of degree n can thus consist of for all integers 0 i < j n 1 . ≤ ≤ − } changing variables to reduce to an equation q (2) Circ0(n) consisting of those with zero coefficient monomial of degree n 1 and then finding a traceless circulant matrix−V V Circ(n) with distinct eigenvalues . { ∈ } that adheres to q. The eigenvalues of V are the 0 (3) M consisting of those roots of pV q and can be readily computed using the representer= P of V . In this program p M with distinct roots . V { ∈ } we seem to be replacing the difficult problem of Each of the subspaces defined is open and dense solving a monic polynomial equation of degree in its respective ambient spaces. It is quite obvious n by the more difficult problem of solving n 1 that nonlinear equations in n 1 variables. However,− 0 0 − : Pn 1 M because of the symmetries present in the latter − → is a complex analytic bijection. An explicit form set of equations, they may be easier to handle. Λ for the inverse to this map would provide an Cubics. We illustrate how this works for cubics algorithm for solving equations of all degrees. by finding a circulant matrix V circ (0,a,b) of = { } Remark 16. We know that zero trace that adheres to 0 0 0 3 Pn 1 Circ (n) M . q(Y ) Y αY β. − ≅ ≅ = + + Each of these spaces is defined analytically. How- We need to find any traceless 3 3 circulant matrix × ever, the last one has an alternate algebraic char- V that adheres to q. Evaluating the representer 2 j 2πı acterization. Let p denote the derivative of p. The PV (Y ) aY bY at y e 3 , j 0, 1, 2, will then ′ = + = = set M0 can be described as yield the roots of q. By formula (5) for detcirc (0,a,b) , we see that p M : deggcd(p, p′) 0 . { } { ∈ = } 3 3 3 pV (Y ) det (Y I V) Y 3abY (a b ), Thus, solving general equations can be reduced by = − = − − + an algebraic procedure to solving equations with and so 3ab α, distinct roots, for the calculation of p is quite al- ′ a3 b3 = −β. gebraic, and so is the calculation of d gcd(p, p′) + = − via the division algorithm. The polynomial= p has It then follows that d 1 1 no multiple roots. 3 3 3 3 β β2 4α β β2 4α The problem encountered above is of funda- a − ± + 27 , b − ∓ + 27 mental importance and quite difficult in general. = 2 = 2 372 Notices of the AMS Volume 59, Number 3 (we are free to choose any consistent set of values Proof. Compute for 0 l < n, ≤ since we need only one representer), and the roots n n 1 d 1 d 1 − − − of q are given by (the values of PV at the three li l(dj i) λl ǫ vi ǫ + vdj i + cube roots of unity) = i 0 = j 0 i 0 = = = n 1 r a b, d d 1 1 − − = +2πı 2πı ldj li 3 2 3 ǫ ǫ vi r2 ae be , = = 2 2πı+ 2πı j 0 i 0 r3 ae 3 be 3 . = = = + d 1 1 ǫnl − ǫli v , − dl i Quartics. In order to find the roots of the polyno- = 1 e i 0 − = mial provided dl is not a multiple of n. In particular, 4 2 n q(Y ) Y βY γY δ, λl 0 for 1 l < d . In general there are n = + + + d integers= l such≤ that 0 Proof. If all the vi ’s are equal, then all the eigen- Proposition 19. Let d n, 1 d < n, and assume values λl of V , except possibly λ0, are equal to n | ≤ n that the vector v C consists of identical blocks zero. We already know that the vanishing of one ∈ d (that is, vi d vi for all i, where indices are cal- λl implies that det V 0. Conversely, assume that + = = culated mod n). Then λl 0 whenever dl is not a det V 0 and that λ 0. Then λ 0 for some = 0 l multiple of n, and V circ v is singular of nullity positive= integer l < n. = Consider the= field exten- = { } n d. sion Q[ǫ] and the automorphism A of this field ≥ − March 2012 Notices of the AMS 373 2 induced by sending ǫ ֏ ǫ (A fixes Q, of course). λ 1, it follows from (8) that the zk (vk 1 | | =k = − − Since n is prime, A generates a cyclic group of vk)λ , k 1,...,n 1, satisfy (9), and thus for = − k automorphisms of Q[ǫ] of order n 1 that acts each k either vk 1 vk or λ 1. The latter holds − − = = transitively on the primitive nth roots of unity: only if λ is actually a dth root of unity, for some 2 n 1 ǫ, ǫ ,...,ǫ − . Hence λl 0 implies that λk 0 divisor d 2 of n, while k is a multiple of d, and { } = = ≥ for all integers k with 1 k n 1. It remains the conclusions of the theorem now follow easily, ≤ ≤ − to show that all the vi’s are equal. Consider the for we may choose the smallest positive integer d (n 1) n matrix such that λd 1. Then d 2, d n and λk 1 for − × 2 n 1 = ≥ | = n 1 ǫ ǫ ǫ − 1 k n if and only if k d, 2d,... or n d d. It 2 4 2(n 1) ≤ ≤ = = 1 ǫ ǫ ǫ − follows that vk vk 1 vk (d 1). = − = = − − ...... . . . . . The next result deals with circulant matrices 2 1 ǫn 1 ǫ2(n 1) ǫ(n 1) − − − whose entries are the same nonzero complex ± (essentially the matrix E in the section “Determi- number. nants and Eigenvalues” with the first row deleted). Proposition 25. If V circ (v0,...,vn 1) Since it has rank n 1, this matrix, when viewed = { − } ∈ Circ(n) has entries in 1 , and 0 < k j vj −Cn Cn 1 {± } = |{ | = as a linear map from to − , must have a one- 1 n k, then dimensional kernel. This kernel is spanned by the }| ≤ − n (a) λ0 0 if and only if k . vector (1, 1,..., 1). The conclusion follows. = = 2 (b) For 0 374 Notices of the AMS Volume 59, Number 3 Proof. By Theorem 6, we have that of polynomials in F[x1,...,xk]. Given any field n 1 extension E of F, we can talk about the locus of − n lj these polynomials in the affine or projective space λl ǫ , = j over the extension field E. These will define the j 0 = E-points of the variety V, a set which we denote by and the binomial expansion yields (a). We obtain l n V(E). This brings about some additional structure that λl 0 if and only if (1 ǫ ) 1, and so = +2πl = 1 to the F-varieties V, which we can think of as a 1 ǫl 1 if and only if cos , a state- n 2 functor from the category of field extensions of | + | = l = −1 l 2 ment equivalent to the condition or . F and their morphisms to a suitable category of n = 3 n = 3 This proves (b). Part (c) follows readily since the sets and morphisms, with the functor mapping an conditions making λl 0 are equivalent to n be- extension E of F to the set V(E) of E-points of the = ing divisible by 2 and 3, respectively, and λl being variety. Using restrictions when possible, we may zero exactly for the two values of l satisfying the also use this idea in the opposite direction and condition in (b). find the points of a variety with coordinates in a subring of F when the variety in question is defined The Geometry of Circ(n) by polynomials whose coefficients are elements of Let k be a positive integer. The affine k-space over the subring. This idea applied to Circ(n) takes us k k C is C , often denoted by AC. The maximal ideals to a rather interesting situation. in the polynomial ring C[x1,...,xk] correspond Given a variety over C cut out by polynomi- k k to elements of C , with a (a1,...,ak) C als with coefficients in Z, we can use the natural = ∈ corresponding to the ideal in C[x1,...,xk] given inclusion Z ֓ C to look at the Z-points of the by the kernel of the evaluation homomorphism variety and the restricted ring of global functions. p ֏ p(a). An affine variety V Ck is an irreduc- In the case of Circ(n), the restricted ring of global ible component of the zero locus⊂ of a collection functions is Z[X]/(Xn 1), and, remarkably, the − of polynomials p1,...,pl in C[x1,...,xk]. The ideal set of prime ideals, or spectrum, of this latter ring IV (p1,...,pl ) C[x1,...,xk] of functions van- is related to a variety defined over a field with one ishing= on V is⊂ prime, and under the above element, a mythical object denoted in the literature identification the points of V are in one-to-one by F1. We elaborate on this connection. It derives correspondence with the set of maximal ideals of from analogies between regular combinatorial ar- guments and combinatorics over the finite field F the ring O(V) C[x1,...,xk]/IV, a ring without q = with q elements (q a power of a prime). zero divisors. We say that V is cut out by p1,...,pl , The number of bases of the F -vector space has ideal IV, and ring of global functions O(V). q k(k 1) k 2− k Theorem 4 realizes Circ(n) as the ring of global Fq is given by q (q 1) [k]q !, where [k]q 2 k 1 − = functions of the variety given by the nth roots of 1 q q q − , and where the q-factorial is + + + + unity in C. defined by [k]q ! [1]q [2]q [k]q . Similarly, the k k = Complex projective k-space P PC is the set number of linearly independent j-element subsets = k 1 j(j 1) of one-dimensional subspaces of C . A point − k + is equal to q 2 (q 1) [k]q !/[k j]q !, and for k − − x P is usually written as a homogeneous vector j k, the number of subspaces of Fk of dimension ∈ ≤ q [x0 : ... : xk], by which is meant the complex line j is given by k 1 spanned by (x0,...,xk) C + 0 . ∈ \{ } k [k]q ! A nonconstant polynomial f C[x0,...,xk] , ∈k P j = [k j]q ![j]q does not descend to a function on . However, if q − f is a homogeneous polynomial of degree d, we an expression that makes perfect sense when k can talk about the zeroes of f in P because we q 1, in which case we obtain the usual binomial. have the relation f (λx ,...,λx ) λdf (x ,...,x ), = 0 k 0 k The idea of the mysterious one element field F1 = k for all λ C 0 .A projective variety V P is emerges [12], and we see that the number of ∈ \{ } ⊂ an irreducible component of the zero locus of a F1-points of projective space—that is to say, the finite collection of homogeneous polynomials. n number of 1-dimensional subspaces of F1 —must If we replace the role of C in the above discus- be equal to n. Speculating on this basis, we are led sion by an arbitraryfield F, we obtain the notions of to define a vector space over F1 simply as a set, a k k-dimensional affine space AF and k-dimensional subspace simply as a subset, and the dimensions k projective space PF over F, respectively. Polyno- of these simply as the cardinality of the said sets. k mials in F[x1,...,xk] define affine varieties in AF, Some relationships between properties of while homogeneous polynomials define projective Circ(n) and that of algebraic geometry over F1 k varieties in PF. These spaces are usually stud- now follow. We think of the group of points ied for algebraically closed F, but the definitions of SL(n, F1) as the symmetric group Sn on n are valid for more general fields, and we work letters and that these n-letters are the F1-points V Pn 1 in this extended context. Let be an affine or of the projective space F1− . A “variety X over projective variety over F, the zero locus of a set F1” should have as extension to the scalars Z, a March 2012 Notices of the AMS 375 scheme XZ of finite type over Z, and the points of coincides, up to row permutations, with the X should be a finite subset of the set of points arbitrary circulant matrix in XZ. Going further, in developing algebraic V circ (v0,v1,...,vn 1) . = { − } geometry over F1, some [13] propose replacing the 2n 2 The intersection C is the image in P − of the notion played by an ordinary commutative ring ∩ 1 points whose coordinates [z0 : z1] P satisfy the by that of a commutative, associative, and unitary n 2 n 3 n 2 ∈ n n equations (z0 − ,zΛ0 − z1,...,z1 − ) (z0 z1 ) 0 or, monoid M to obtain Spec (M F1 Z) Spec Z[M]. n n − i= 1 ⊗ = equivalently, z0 z1 0. The point [1 : ǫ ] P In particular, they define the finite extension F n 1 gets mapped to the− point= ∈ of degree n as the monoid Z/nZ, and its spectrum i 2i (n 1)i after lifting it to Z becomes pi [1: ǫ : ǫ : : ǫ − ], 0 i n 1, = ≤ ≤ − n and so the restriction of Ik to vanishes on Spec (F1n F Z) Spec (Z[X]/(X 1)) . ⊗ 1 = − span(pi1 , pi2 ,...,pik 1 ). Thus, the algebra of circulant matrices with integer Λ − i1,i2,...,ik 1 0,...,n 1 coefficients is the ring of global functions of the − ∈{ − } In particular, the determinant of the circulant spectrum of the field extension F n of degree n 1 matrix V vanishes on the union of the n distinct after lifting it to Z. hyperplanes The Rational Normal Curves Connection span(p0, p1,..., pi ,...,pn 1), − i 0,...,n 1 Theorem 6 has an elaborate proof that is more ∈{ − } geometric in nature and longer than the proof by where the symbol pi indicates that pi does not 2n 2 calculation given above. We outline its details. appear. The union of these n hyperplanes in P − d The rational normal curve Cd P of degree d is a degree n subvariety of codimension 1, and ⊂ 1 d is defined to be the image of the map P P thus any degree n polynomial vanishing on it → given by must be its defining equation, up to a scalar factor (because for any hypersurface, its defining ideal is d d 1 d 1 d [z0 : z1] ֏ [z : z − z1 : : z0z − : z ] 0 0 1 1 generated by one element and the degree of the [Z0 : : Zd]. hypersurface is the degree of this element). We = deduce that det(V) factors as in the statement of It is the common zero locus of the polynomials Theorem 6. pij Zi Zj Zi 1Zj 1 for 1 i j d 1. The ideal Similarly, though the argument is slightly more = − − + ≤ ≤ ≤ − of Cd , I(Cd ) f C[Z0,...,Zd] f 0 on Cd is involved, we can show also that for all k ={ ∈ | ≡ } generated by this set of polynomials. 2,...,n , the ideal of k k-minors of the generic∈ { } × We view v0,...,vn 1,...,v2n 2 as a set of circulant matrix V defines the (reduced) union of { − − } 2n 1 independent variables and consider the (k 2)-planes − matrix with constant antidiagonals given by − span(pi1 , pi2 ,...,pik 1 ) − v0 v1 vn 2 vn 1 i1,i2,...,ik 1 0,...,n 1 − − − ∈{ − } v1 v2 vn 1 vn (in contrast with the case of the generic catalec- − ...... ticant matrix, where all ideals of minors are M . . . . . = prime). vn 2 vn 1 v2n 4 v2n 3 − − − − v v v v n 1 n 2n 3 2n 2 − − − Other Connections—Toeplitz Operators as an n n catalecticant or Hankel matrix. Its 2 2- We end by discussing briefly a relation between × × minors define the idealofthe rationalnormalcurve circulant and Toeplitz matrices. The interested 2n 2 C C2n 2 P − of degree 2n 2. reader may consult [6] for more information about = − ⊂ − The other ideals of minors of M also have geo- the connection. metric significance. Since the sum of m matrices Let t n 1,...,t0,...,tn 1 be a collection of { − + − } of rank one has rank at most m, the ideal I of 2n 1 complex numbers. An n n matrix T [tkj ] k − × = k k-minors of M, k 2,...,n , vanishes on is said to be Toeplitz if tkj tk j . Thus, a Toeplitz × ∈ { } = − the union of the (k 1)-secant (k 2)-planes to matrix T is a square matrix of the form − P2n −2 the rational normal curve C − . The ideal Ik t0 t 1 t (n 2) t (n 1) ⊂ − − − − − defines the (reduced) locus of these (k 1)-secant t1 t0 t (n 3) t (n 2) − − − − − (k 2)-planes to C [14] (see [2] for a modern ...... − T Tn . . . . . . proof). = = tn 2 tn 3 t0 t 1 The restriction of M to the (n 1)-dimensional − − − − t t t t linear subspace n 1 n 2 1 0 − − These matrices have a rich theory, and they relate vn v0 vn 1 v1 v2n 2 vn 2 0 ={ − = + − = = − − − = } naturally to the circulant ones we study here. If 2n 2 P − we have tk t (n k) tk n, then as a special case Λ ⊂ = − − = − 376 Notices of the AMS Volume 59, Number 3 Tn is circulant. We use both classes of matrices in by πnϕ. The matrix Vn(πnϕ) is also Toeplitz, and a proof of a celebrated spectral theorem to show its Toeplitz’s coefficients are determined solely by the depth of their interconnection. t n 1,...,tn 1 . The matrices Vn(ϕ) and Vn(πnϕ) { − + − } Let ϕ be a smooth real-valued function on are asymptotically equivalent, and a simple L2- the unit circle with Fourier coefficients ϕˆ j argument of Fourier series shows that the latter 2π = ıjθ is asymptotically equivalent to Tn(ϕ), and so also 0 e− ϕ(θ)dθ, and consider the Toeplitz matrix the former. Tn(ϕ) (ϕˆ i j ), 0 i, j n 1. The renowned = − ≤ ≤ − Szegö theorem [7] asserts that if f is a continuous Arbitrary powers of Tn and Vn have asymp- function on C, then totically equivalent eigenvalues, and the general Szegö’s theorem follows by applying Weierstrass’s 1 1 (10) lim f (λ) f (ϕ(eiθ ))dθ. polynomial approximation to f . n n = 2π S1 →∞ λ specTn(ϕ) It is of practical significance that Vn(πnϕ) ∈ We sketch a classical argument leading to its proof. encodes finite-dimensional information of the 1 Fourier expansion of ϕ and spectral information Given a double sequence tk k+∞ C in l (and hence also in l2), let ϕ {be} the=−∞L1⊂-function on the zeroth order pseudodifferential operator πnMϕπn, where Mϕ is the multiplication by ϕ whose Fourier coefficients are the tj ’s. We form the operator. sequence of Toeplitz matrices Tn(ϕ) Tn n+∞1, { = } = where Tn is defined, as above, by t n 1,...,tn 1 , (n) { − + − } References and denote by τl , l 0, 1,...,n 1 its eigenval- ues. The T ’s are Hermitian= if and− only if ϕ is [1] P. J. Davis, Circulant Matrices, AMS Chelsea n Publishing, 1994. real-valued. We study the asymptotic distribution [2] D. Eisenbud, Linear sections of determinantal n 1 varieties, Amer. J. Math. 110 (1988), 541–575. 1 − (n) (11) lim τl , [3] H. M. Farkas and I. Kra, Theta Constants, Riemann n n →+∞ l 0 Surfaces and the Modular Group, Graduate Stud- = the case of f (x) x in Szegö’s identity (10). ies in Mathematics, vol. 37, American Mathematical = Society, 2001. We introduce the circulant matrix Vn(ϕ) (n) (n) = [4] P. A. Fuhrmann, A Polynomial Approach to Linear circ (v0 ,...,vn 1) , where { − } Algebra, Universitext, Springer, 1996. n 1 [5] D. Geller, I. Kra, S. Popescu, and S. Simanca, (n) 1 − 2πj 2πjk (12) v ϕ e n i . On circulant matrices, preprint, 2002 (pdf at k = n n j 0 http://www.math.sunysb.edu/~sorin/eprints/ = circulant.pdf). For fixed k, this is the truncated Riemann sum [6] R. M. Gray, Toeplitz and Circulant Matrices: A Re- approximation to the integral yielding t k, and view (Foundations and Trends in Communications 1 (n) − since ϕ L ,we have vk t k. ByTheorem6, the and Information Theory), NOW, 2005. ∈ → − (n) l [7] U. Grenander and G. Szegö, Toeplitz Forms and ordered eigenvalues of Vn(ϕ) are λl ϕ 2π , = n Their Applications, University of California Press, l 0,...,n 1, and so, using Riemann sums to 1958. approximate= − the integral of the mth power of ϕ, [8] D. Kalman and J. E. White, Polynomial equations m N, we conclude that ∈ and circulant matrices, Amer. Math. Monthly 108 n 1 1π (2001), 821–840. − 1 (n) m 1 m (13) lim (λl ) ϕ(θ) dθ. [9] I. Kra, Product Identities for θ-Constants and n n = 2π →∞ l 0 0 θ-Constant Derivatives, in preparation. = This relates the average of ϕ to the asymptotic [10] T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, J. Algebra 224 (2000), 91–109. Also distributions of the eigenvalues of V . The special n arXiv:math/9511209, November 1995. case of Szegö’s theorem above is now within reach. [11] H. R. Parks and D. C. Wills, An elementary calcula- If we can prove that the two sequences of n n × tion of the dihedral angle of the regular n-simplex, matrices Tn and Vn are asymptotically equiv- Amer. Math. Monthly 109 (2002), 756–758. { } { } alent in the sense that limn Tn Vn 0, [12] J. Tits, Sur les Analogues Algébriques des Groupes where V is the Hilbert-Schmidt→+∞ || norm− of the|| = op- Semi-Simples Complexes, Colloque d’algèbre erator V , then their eigenvalues are asymptotically supérieure, Bruxelles, 1956, pp. 261–289, Centre equivalent in the sense that Belge de Recherches Mathématiques, Louvain; Librairie Gauthier-Villars, 1957, Paris. n 1 1 − (n) (n) [13] B. Töen and M. Vaquié, Au-dessus de SpecZ, lim (τl λl ) 0, arXiv:math/0509684, October 2007. n n − = →+∞ l 0 [14] R. Wakerling, On the loci of the (K 1)-secant = + and so (11) equals (13) for m 1. It is convenient K-spaces of a curve in r-space, Ph.D. thesis, Berkeley, to do this by introducing the= auxiliary circu- 1939. (n) (n) [15] E. W. Weisstein, Circulant Matrix, from lant matrix Vn(πnϕ) circ (v˜0 ,..., v˜n 1) of = { n 1 − } ijθ MathWorld—A Wolfram Web Resource, http:// the truncated Fourier series πnϕ j + n 1 tj e , mathworld.wolfram.com/CirculantMatrix. (n) = =− + where v˜k is given by(12) with the role of ϕ played html. March 2012 Notices of the AMS 377