The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs

The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs tài liệu, giáo án, bài...

Forum Math 26 (2014), 141 – 175 DOI 10.1515 / FORM.2011.155 Forum Mathematicum © de Gruyter 2014 The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs Le Anh Vinh Communicated by Christopher D Sogge Abstract In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods Roughly speaking, we show that almost all systems of norm, bilinear or quadratic equations over finite fields are solvable in any large subset of vector spaces over finite fields Keywords Bilinear equations, quadratic equations, spectral graphs, finite fields 2010 Mathematics Subject Classification 11L40, 11T30 Introduction The main purpose of this paper is to give a unified proof of several results on the solvability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods We will see that after appropriate graph theoretic results are developed, many old and new results immediately follows In this section, we discuss the motivation and background results for our work Let Fq denote a finite field with q elements, where q, a power of an odd prime, is viewed as an asymptotic parameter For E Fqd (d 2), the finite analogue of the classical Erd˝os distance problem is to determine the smallest possible cardinality of the set .E/ D ¹kx yk D x1 y1 /2 C C xd yd /2 W x; y Eº Fq : The first non-trivial result on the Erd˝os distance problem in vector spaces over finite fields is due to Bourgain, Katz, and Tao ([9]), who showed that if q is a prime, q Á (mod 4), then for every " > and E Fq2 with jEj Ä C" q , there exists Cı ı > such that j.E/j Cı q for some constants C" ; Cı The relationship between " and ı in their arguments, however, is difficult to determine In addition, it is quite subtle to go up to higher dimensional cases with these arguments Iosevich and Rudnev ([19]) used Fourier analytic methods to show that there are absolute constants c1 ; c2 > such that for any odd prime power q and any set E Fqd 142 of cardinality jEj L A Vinh c1 q d=2 , we have j.E/j ° ± d c q; q jEj : (1.1) Iosevich and Rudnev reformulated the question in analogy with the Falconer distance problem: how large does E Fqd , d 2, needed to be ensure that .E/ contains a positive proportion of the elements of Fq The above result implies that d C1 if jEj 2q , then .E/ D Fq directly in line with Falconer’s result in Euclidean setting that for a set E with Hausdorff dimension greater than d C 1/=2 the distance set is of positive measure At first, it seemed reasonable that the exponent d C 1/=2 may be improvable, in line with the Falconer distance conjecture described above However, Hart, Iosevich, Koh and Rudnev discovered in [12] that the arithmetic of the problem makes the exponent d C 1/=2 best possible in odd dimensions, at least in general fields In even dimensions it is still possible that the correct exponent is d=2, in analogy with the Euclidean case In [10], Chapman et al took a first step in this direction by showing that if a set E Fq2 satisfies 4=3 jEj q , then j.E/j cq This is in line with Wolff’s result for the Falconer conjecture in the plane which says that the Lebesgue measure of the set of distances determined by a subset of the plane of Hausdorff dimension greater than 4=3 is positive In [30], the author gave another proof of (1.1) using the graph theoretic method (see also [37] for a similar proof) The (common) main step of these proofs is to estimate the number of occurrences of a fixed distance It was shown that for a fixed distance, given that the point set is large, the number of occurrences of any fixed distance is close to the expected number This implies that there are many distinct distances occur in a large point set In the case of real number field, most of the known results, however, are actually proved in a stronger form In order to show that there are at least g.n/ distinct distances determined by an n-point set in the plane, one usually proves that for any n-point set P , there exists a point p P that determines at least g.n/ distinct distances to P Chapman et al ([10]) obtained an analogous result in the finite field setting They also proved a similar result for the pinned dot product sets …y E/ D ¹x y W x Eº In this paper, we will derive these results using spectral graph methods A classical result due to Furstenberg, Katznelson and Weiss ([13]) states that if E R2 of positive upper Lebesgue density, then for any ı > 0, the ı-neighborhood of E contains a congruent copy of a sufficiently large dilate of every threepoint configuration An example of Bourgain ([7]) showed that it is not possible to replace the thickened set Eı by E for arbitrary three-point configurations In the case of k-simplex, that is the k C points spanning a k-dimensional subspace, Bourgain ([7]), using Fourier analytic techniques, showed that a set E of positive Sovability of equations over finite fields 143 upper Lebesgue density always contains a sufficiently large dilate of every nondegenerate k-point configuration where k < d In the case k D d , the problem still remains open Using Fourier analytic method, Akos Magyar ([23, 24]) considered this problem over the integer lattice Zd He showed that a set of positive density will contain a congruent copy of every large dilate of a non-degenerate k-simplex where d > 2k C Hart and Iosevich ([18]) made the first investigation in an analog of this question in finite field geometries Let Pk denote a k-simplex Given another k-simplex Pk0 , we say Pk Pk0 if there exist Fqd , and O SOd Fq /, the set of d -by-d orthogonal matrices over Fq , such that Pk0 D O.Pk / C Under this equivalent relation, Hart and Iosevich ([18]) observed that one may specify a simplex by the kC1 distances determined by its vertices They showed that if E Fqd (d ) is kd k C a set of cardinality jEj & C q kC1 , then E contains a congruent copy of every k-simplices (with the exception of simplices with zero distances) Using graph theoretic method, the author ([35]) showed that the same result holds for d 2k and d jEj q Ck Here, and throughout, X Y means that there exists C > such that X Ä C Y , and X Y means that X D o.Y / Note that serious difficulties arise when the size of simplex is sufficiently large with respect to the ambient dimension Even in the case of triangles, the result in [35] is only non-trivial for d Covert, Hart, Iosevich, and Uriarte-Tuero ([11]) addressed the case of triangles in plane over finite fields They showed that if E has density for some C q 1=2 Ä Ä with a sufficiently large constant C > 0, then the set of triangles determined by E, up to congruence, has density c In [36], the author studied the remaining case: triangles in three-dimensional vector spaces over finite fields Using a combination of graph theory method and Fourier analytic techniques, the d C2 author showed that if E Fqd (d 3) of cardinality jEj & C q , the set of triangles, up to congruence, has density greater than c Using Fourier analytic techniques, Chapman et al ([10]) extended this result to higher dimensional cases d Ck More precisely, they showed that if jEj & q (d k), then the set of k-simplices, up to congruence, has density greater than c They also obtained a stronger result when E is a subset of the d -dimensional unit sphere S d D ¹x Fqd W kxk D 1º: In particular, it was proven ([10, Theorem 2.15]) that if E S d of cardinality d Ck jEj & q , then E contains a congruent copy of a positive proportion of all k-simplices In this paper, we will obtain similar results in a more general setting Let Q be a non-degenerate quadratic form on Fqd The Q-distance between two points x; y Fqd is defined by Q.x y) We consider the system L of k2 equations Q.x i xj / D ij ; x i E; i D 1; : : : ; k; (1.2) 144 L A Vinh over Fqd , with variables from arbitrary set E if jEj q d Ck Fqd We show that , then the system (1.2) is solvable for all ij Fq , k if jEj q d Ck/=2 , then that system 1.2 is solvable for at least o.1//q / possible choices of ij Fq A related question that has recently received attention is the following Letting A Fq , how large does A need to be to ensure that Fq A A C C A A (d times) Bourgain ([8]) showed that if A Fq of cardinality jAj C q 3=4 , then A A C A A C A A D Fq Glibichuk and Konyagin ([16]) proved in the case p of prime fields Zp that for d D 8, one can take jAj > q Glibichuk ([15]) then extended this result to arbitrary finite fields Note that this question can be stated in a more general setting Letting E Fqd , how large does E need to be to ensure that the equation x y D ; x; y E; is solvable for any given Fq Hart and Iosevich ([18]), using exponential sums, showed that one can take jEj > q d C1/=2 for any d In this paper, we will give another proof of this result using spectral graph methods In analogy with the study of simplices in vector spaces over finite fields, the author ([31]) studied the sovability of systems of bilinear equations over finite fields More precisely, for any non-degenerate bilinear form B ; / in Fqd , we consider the following system of l Ä k2 equations: B.ai ; aj / D ij ; E; i D 1; : : : ; k; (1.3) over Fqd , with variables from an arbitrary set E Fqd Using character sum machinery and methods from graph theory, the author ([31]) showed that if each varid able in the system (1.3) appears in at most t Ä k equations and jEj q Ct , then for any ij Fq , the system (1.3) has C o.1//q l jEjk solutions Again, serious difficulties arise when the number of equations that each variable involves is sufficiently large with respect to the ambient dimension In particular, that result is only non-trivial in the range of d 2t In the case of three variables and three equations, the author also proved ([31, Theorem 1.4]) that the system (1.3) is solvd C2 able for o.1//q triples 12 ; 23 ; 31 / Fq /3 if jEj q In this paper, we will extend this result to systems with many variables More precisely, we will d Ck show that if E Fqd is a set of cardinality jEj q , then the system (1.3) k of all k2 equations is solvable for o.1//q / possible choices of ij Fq , Ä i < j Ä k We remark here that one can also obtain this result using Fourier analytic methods (for example, using [10, Theorem 2.14] instead of [10, Theorem 2.12] in the Sovability of equations over finite fields 145 proof of [10, Theorem 2.13]) However, techniques involved in difference problems are considerable in Fourier analytic proofs The main advantage of our approach is that we can obtain all the aforementioned results at once, after computing the eigenvalues of appropriate graphs We will also demonstrate our method by some related results on norm equations and sum-product equations over finite fields 2.1 Statement of results Subgraphs in n; d; /-graphs For a graph G, let k be the eigenvalues of its adjacency matrix The quantity G/ D max¹ ; n º is called the second eigenvalue of G A graph G D V; E/ is called an n; d; /-graph if it is d -regular, has n vertices, and the second eigenvalue of G is at most It is well known (see [4, Chapter 9] for more details) that if is much smaller than the degree d , then G has certain random-like properties Noga Alon ([21, Theorem 4.10]) proved that every large subset of the set of vertices of n; d; /-graphs contains the “correct” number of copies of any fixed sparse graph Theorem 2.1 ([21, Theorem 4.10]) Let H be a fixed graph with r edges, s vertices, and maximum degree , and let G D V; E/ be an n; d; /-graph where d Ä 0:9n Let m < n satisfy m n=d / Then, for every subset V V of cardinality m, the number of (not necessarily induced) copies of H in V is  Ãr ms d C o.1// : j Aut.H /j n If we are only interested in the existence of one copy of H , then one can sometimes improve the conditions on d and in Theorem 2.1 The first result of this paper is an improvement of the conditions on d and in Theorem 2.1 for complete bipartite graphs Let G G be the bipartite graph with two identical vertex parts V G/ and V G/ Two vertices u and v in two different parts are connected by an edge if and only if they are connected by an edge in G For any two subsets U1 ; U2 V G/, let GŒU1 ; U2  be the induced bipartite subgraph of G G on U1 U2 Theorem 2.2 For any t For every subsets U1 ; U2 s and t V with jU1 jjU2 j 2, let G D V; E/ be an n; d; /-graph n=d /tCs ; 146 L A Vinh the induced subgraph GŒU1 ; U2  contains jU1 js jU2 jt C o.1// sŠt Š  Ãst d n copies of Ks;t Note that the bound in Theorem 2.2 is stronger than that in Theorem 2.1 when t > s For small bipartite subgraphs, K2;t , we can further improve the bound in Theorem 2.2 Theorem 2.3 For any t sets U1 ; U2 V with 1, let G D V; E/ be an n; d; /-graph For every subjU1 jjU2 j n=d /t C1 the induced subgraph GŒU1 ; U2  contains jU1 js jU2 jt C o.1// 2Št Š  Ãst d n copies of K2;t In fact, our results could be stated in multi-color versions, which will be more convenient in later applications Suppose that a graph G is edge-colored by a set of finite colors We call G an n; d; )-colored graph if the subgraph of G on each color is an n; d.1 C o.1//; /-graph The following results are multi-color analogues of Theorems 2.1, 2.2 and 2.3 for n; d; /-colored graphs Theorem 2.4 Let H be a fixed edge-colored graph with r edges, s vertices, and maximum degree , and let G D V; E/ be an n; d; /-colored graph, where d Ä 0:9n Let m < n satisfy m n=d / For every subset V V of cardi0 nality m, the number of (not necessarily induced) copies of H in V is  Ãr ms d C o.1// : j Aut.H /j n Theorem 2.5 For any t 2, let H be a fixed edge-colored complete bipartite graph Ks;t with s Ä t Let G D V; E/ be an n; d; /-colored graph For every subsets U1 ; U2 V with jU1 jjU2 j n=d /tCs ; the induced subgraph GŒU1 ; U2  contains jU1 js jU2 jt C o.1// Aut.H / copies of H  Ãst d n Sovability of equations over finite fields 147 Theorem 2.6 For any t 1, let H be a fixed edge-colored complete bipartite graph K2;t , and let G D V; E/ be an n; d; /-colored graph For every subsets U1 ; U2 V with jU1 jjU2 j n=d /t C1 ; the induced subgraph GŒU1 ; U2  contains C o.1// jU1 j2 jU2 jt Aut.H /  Ã2t d n copies of H The proof of Theorem 2.4 is similar to that of [21, Theorem 4.10], the proofs of Theorem 2.5 and Theorem 2.6 are similar to the proofs of Theorem 2.2 and Theorem 2.3, respectively To simplify the notation, we will only present the proofs of single-color results Note that going from single-color formulations (Theorems 2.1, 2.2 and 2.3) to multi-color formulations (Theorems 2.4, 2.5 and 2.6) is just a matter of inserting different letters in a couple of places Although we cannot improve the conditions on d and in Theorem 2.4 (or equivalently Theorem 2.1), we will show that if the number of colors is large, under a weaker condition, any large induced subgraph of an n; d; /-color graphs contains almost all possible colorings of small complete subgraphs Theorem 2.7 For any t 2, let G D V; E/ be an n; d; /-colored graph, and let m < n satisfy m n=d /t=2 Suppose that the color set C has cardinality jCj D o.1//n=d Then for every subset U V with cardinality m, the int duced subgraph of G on U contains at least o.1//jCj.2/ possible colorings of K t The results above could also be considered as a contribution to the fast-developing comprehensive study of graph theoretical properties of n; d; /-graphs, which has recently attracted lots of attention both in combinatorics and theoretical computer science For a recent survey about these fascinating graphs and their properties, we refer the interested reader to the paper of Krivelevich and Sudakov ([21]) 2.2 Norms in sum sets, pinned norms, and norm equations Let Fq be a finite field with q D p d elements We denote by FN an algebraic closure of Fq , and by Fq n FN the unique extension of the degree n of F for n The n extension Fq =Fq is a Galois extension, with Galois group Gn canonically isomorphic to Z=Zn , the isomorphism being the map Z=Zn ! Gn defined by 7! , 148 L A Vinh where is the Frobenius automorphism of Fq n given by X/ D X q Associated to the extension Fq n =Fq , the norm map N D NFqn =Fq W Fq n ! Fq n is defined by N.X / D nY1 i i D0 X / D nY1 i Xq D X qn q : i D0 The equation N.X / D , for a fixed Fq , is important in number theory Because the extension Fq n =Fq is separable, the equation N.X / D is always solvable with X Fq n for any Fq We are interested in the solvability of this equation when X is in a sum set of two large subsets of Fq n More precisely, we have the following result Theorem 2.8 Let Fq and A; B  Fq n , n Suppose that jAjjBj Then the equation N.X C Y / D is solvable in X A, Y B For any A; B q nC2 Fq n , define by N.A C B/ the norm set of the sum set, i.e N.A C B/ D ¹N.X C Y / W X A; Y Bº: Theorem 2.8 says that if jAjjBj q nC2 , then Fq N A C B/ We will show that under a slightly stronger condition, one can always find many elements X A such that the pinned norm set NX B/, which is defined by NX B/ D ¹N.X C Y / W Y Bº; contains almost all elements in Fq Theorem 2.9 Let A; B  Fq n , n Suppose that A; B satisfy jAj jBj and jAjjBj q nC2 Then there exists a subset A0 of A with cardinality jA0 j & jAj such that for every X A0 , the equation N.X C Y / D is solvable in Y B for at least o.1//q values of F We also obtain the following results on the solvability of systems of norm equations over finite fields Theorem 2.10 Let A  Fq n , n Consider the systems L of l Ä tions N.Xi C Xj / D ij ; Xi A; i D 1; : : : ; t: t norm equa(2.1) Suppose that each variable appears in at most k Ä t equations, and suppose that jAj q n=2Ct Then for any ij Fq , the above system has C o.1//q solutions l jAjt Sovability of equations over finite fields 149 Theorem 2.11 Let A  Fq n , n Consider the system (2.1) with 2t equations t If jAj q nCt/=2 , then that system is solvable for at least o.1//q 2/ choices of ij Fq , Ä i < j Ä t 2.3 Dot product set and system of bilinear equations Let E; F Fqd D Fq Fq , d For any non-degenerate bilinear form d B ; / on Fq , define the product set of E and F with respect to B by B.E; F / D ¹B.x; y/ W x E; y F º: Hart and Iosevich ([18]), using character sum machinery, proved the following result Theorem 2.12 ([18, Theorem 2.1]) Let E; F Then Fq  B.E; F / Let E Fqd , d Fqd Suppose that jEjjF j q d C1 Define the pinned product set by By E/ D ¹B.x; y/ W x Eº: Chapman et al ([10]) obtained the following result using Fourier analytic methods Theorem 2.13 ([10, Theorem 2.4]) Let E Fqd , d 2, be a set of cardinality d C1/=2 jEj q Then there exists a subset E E of cardinality jE j & jEj such that for every y E , one has jBy E/j > q=2 Note that [18, Theorem 2.1] and [10, Theorem 2.4] are stated only for the dot product, but their proofs go through for any non-degenerate bilinear form without any essential change As a corollary of our results in Section 2.1, we will give graph theoretic proofs of Theorems 2.12 and 2.13 In fact, we will prove the following result instead of Theorem 2.13 Theorem 2.14 Let E Fqd , d 2, of cardinality jEj q d C1/=2 Then there ex0 ists a subset E E of cardinality jE j D o.1//jEj such that for every y E , one has jBy E/j D o.1//q Note that the proof of [10, Theorem 2.14] also implies Theorem 2.14 and vice versa We, however, relax the condition on jEj q d C1/=2 to jEj q d C1/=2 to simplify our arguments In [31], the author studied the solvability of systems of bilinear equations over finite fields Following the proof of [21, Theorem 4.10], the author proved the following result 150 L A Vinh Theorem 2.15 ([31]) Let E  Fqd , d For any non-degenerate bilinear form B ; / on Fqd , consider the systems L of l Ä 2t bilinear equations B.ai ; aj / D d ij ; A; i D 1; : : : ; t: (2.2) Suppose that jEj q Ct and each variable appears in at most k Ä t equations Then the system (2.2) is solvable for any ij Fq , Ä i < j Ä t As a simple consequence of Theorem 2.7 and the construction of product graph in Section 8, we show that under a weaker condition, say jAj q nCt 1/=2 , the system (2.2) is solvable for almost all possible choices of parameters ij F Theorem 2.16 Let E  Fqd , d 2, of cardinality jEj q nCt 1/=2 For any non-degenerate bilinear form B ; / on Fqd , consider the systems L of 2t bilinear equations B.ai ; aj / D ij ; A; i D 1; : : : ; t: (2.3) Then the above system is solvable for 1 Ä i < j Ä t 2.4 t o.1//q 2/ possible choices of ij F, Sum-product equations In [28], Sárközy proved that if A; B; C ; D are “large” subsets of Zp , more precisely, jAjjBjjCjjDj p , then the sum-product equation a C b D cd (2.4) can be solved with a A; b B; c C and d D Gyarmati and Sárközy [17] generalized this result on the solvability of equation (2.4) to finite fields They also studied the solvability of other (higher degree) algebraic equations with solutions restricted to “large” subsets of Fq Using bounds of multiplicative character sums, Shparlinski [27] extended the class of sets which satisfy this property Furthermore, Garaev [14] considered the equation (2.4) over special sets A; B; C; D to obtain some results on the sum-product problem in finite fields More precisely, he proved the following theorem Theorem 2.17 ([14]) For any A  Fq , we have q jA AjjAj2 jA C Aj jAj3 Ä C qjA AjjAj2 jA C Aj; q which implies that jA C AjjA Aj ³ ² jAj4 qjAj; : q 162 L A Vinh Definition 6.2 Given U Ut D ¹.y1 ; : : : ; y t Ut Á U V , let U Moreover, for each y1 ; : : : ; y t U.y1 ; : : : ; y t For any two sets U1 1/ 1/ U, t W y1 ; : : : ; y t 1/ Ut 1, Uº: define D ¹y t W y1 ; : : : ; y t U2 , we say that U1 jU1 j D 1; yt / Define 1; yt / Uº U: U2 if and only if o.1//jU2 j: We can now state a slightly stronger version of Lemma 6.1 Lemma 6.3 For any t and t colors r1 ; : : : ; r t , let G D V; E/ be an n; d; /-colored graph Suppose that the color set C has cardinality jCj D V of cardinality jU j For any subset U U with U o.1//n=d: jU jt , we have X n=d /t=2 and for any Ut D U X U t Iyr11;:::;r ;:::;y t U.y1 ; : : : ; y t // Proof The upper bound is trivial so it suffices to show that X X t Iyr11;:::;r ;:::;y t U.y1 ; : : : ; y t // y1 ;:::;y t /2U t D y1 ;:::;y t /2U t y1 ;:::;y t /2U t r1 ;:::;r t t : jjCj 2C o.1//jU t t : jjCj o.1//n=d colors, so jE.G/j D o.1//n2 =2 Hence, X t S yr11;:::;r ;:::;y t U.y1 ; : : : ; y t // r1 ;:::;r t 2C X y1 ;:::;y t D 1 2C o.1//jU t Note that we have X r1 ;:::;r t 1 X 2U r1 ;:::;r t o.1//jU jt : ;:::;r t Syr11;:::;y t 1 U / jU jt U/ 2C (6.3) 163 Sovability of equations over finite fields On the other hand, it follows from Lemma 5.1 that X X t S yr11;:::;r ;:::;y t U.y1 ; : : : ; y t y1 ;:::;y t /2U t r1 ;:::;r t 2C X Ä y1 ;:::;y t /// X /2U t r1 ;:::;r t t S yr11;:::;r ;:::;y t U //2 2C  Ã2.t t C1 d D C o.1//jU j n D C o.1//jU jt 1 =jCjt 1/ jCjt ; (6.4) since jC j D o.1//n=d The lemma now follows from (6.3), (6.4), and the Cauchy–Schwarz inequality By the pigeon-hole principle, we have an immediate corollary of Lemma 6.3 Corollary 6.4 For any t and t colors r1 ; : : : ; r t , let G D V; E/ be an n; d; /-colored graph Suppose that the color set C has cardinality jCj D U Ut D U jU jt , there exists a subset U.t X r1 ;:::;r t n=d /t=2 and for any V of cardinality jU j For any subset U with U that o.1//n=d: t Iyr11;:::;r ;:::;y t 1 U 1/ U.y1 ; : : : ; y t Ut // with U.t D 1/ o.1//jCjt Ut such 1 2C for every y1 ; : : : ; y t 1/ U.t 1/ This corollary says that for any large set U U t , there exists a large subset t t 1/ t Ut U such that U jU j , and any given t 1/-tuple t 1/ y1 ; : : : ; y t / U is extendable to at least o.1//jCjt types of t 1/stars with roots in U.y1 ; : : : ; y t / The rest of the proof is easy For any U V with cardinality jU j n=d /t=2 , we construct t sets U.t/ ; : : : ; U.1/ inductively t/ as follows Let U D U t D U U Since jU j n=d /t=2 , from Corol.t/ t 1/ lary 6.4, we can choose U U t such that any t 1/-tuple y1 ; : : : ; y t / in U.t 1/ is extendable to at least o.1//jCjt types of t 1/-stars with roots in U.t / y1 ; : : : ; y t / Suppose that we have constructed the sets U.t/ ; : : : ; U.i/ (i 2) such that U.j / U j for i Ä j Ä t 1, and any given j -tuple y1 ; : : : ; yj / U.t 1/ 164 L A Vinh in U.j / is extendable to at least U.j C1/ y1 ; : : : ; yj / Since o.1//jCjj types of j -stars with roots in n=d /t=2 jU j n=d /i=2 ; from Corollary 6.4 again, we can choose U.i i/ 1/ Ui such that U.i 1/ U i , and any given t 1/-tuple y1 ; : : : ; y t / in U.i 1/ is extendable to at least o.1//jCji types of i 1/-stars with roots in U.i/ y1 ; : : : ; yi / Repeat the process until we have constructed U.1/ 2/ U1 U of cardinality jU.1/ j jU j Taking any vertex v in U.1/ , it is clear that we can ext tend from v to at least o.1//jCj.2/ types of edge-colored K t in U t This completes the proof of Theorem 2.7 Projective norm graphs We first recall the construction of the projective norm graphs N Gq;n / in [2] It is possible to get a slightly better version of these ([3]), but this makes no essential difference for our purpose here The construction is the following Let n be an integer and q be an odd prime power Let Fq be the finite field of q elements, and Fq n be the unique extension of degree n of Fq The vertex set of the graph N Gq;n / is the set V D Fq n Two distinct vertices X and Y V are adjacent if and only if N.X C Y / D , where the norm N is defined as in Section 2.2 with an extension N.0/ D For any Fq , the equation N.X/ D has a solution, and if X0 is a given solution, the set of solutions is in one-to-one correspondence with solutions of N.X / D 1, which by Hilbert’s Theorem 90 (or by direct proof) is given by X D Y /Y D Y q for some Y Fq n (see [20] for more details) Hence, all projective norm graphs N Gq;n / ( Fq ) are isomorphic It follows immediately from the definition that N Gq;n 1/ is a regular graph of order q n and valency q n 1/=.q 1/ The eigenvalues of this graph is not hard to compute We present here only a sketch of the proof, which follows the presentation of [2] Let A be the adjacency matrix of N Gq;n / The rows and columns of this matrix are indexed by the ordered pairs of the set Fq n Let be a character of the additive n group of PFq One can2 check (see [2]) that is2an eigenvector of A with eigenvalue j N.c/D1 c/j and all eigenvalues of A are of this form The trivial character is corespondent to the large eigenvalue q t 1/=.q 1/ of A The others can Sovability of equations over finite fields 165 be estimated using Weil’s bound on the character sum (see [26, Theorem 2E (i)]), and the fact that all solutions of N.X / D are all given by X D Y /Y D Yq for some Y Fq n , ˇ X ˇ ˇ ˇ N.c/D1 ˇ ˇ X ˇ ˇ c/ˇˇ D ˇˇ q d d 2Fq n q ˇ ˇ q /ˇˇ < 2/q n=2 < q n=2 : q Therefore, we have the following result Lemma 7.1 For any n a and n q n ; qq 11 ; q n=2 /-graph 7.1 Fq , the projective norm graph N Gq;n / is Proofs of Theorems 2.8, 2.9, 2.10 and 2.11 We are now ready to prove results on norm equations in Section 2.2 For any two set A; B  Fq n and Fq , let e A; B/ D #¹.X; Y / A B W N.X CY / D º It follows from Corollary 3.2 and Lemma 7.1 that ˇ ˇ n p ˇ 1/jAjjBj ˇˇ n=2 ˇe A; B/ q < q jAjjBj: ˇ q 1/q n ˇ Hence we have e A; B/ > if jAjjBj q nC2 This completes the proof of Theorem 2.8 Next, we consider the graph G on the vertex set Fq n Two distinct vertices X and Y are connected by an -colored edge if and only if N.X CY / D Note that we only use q 1/ colors Fq From Lemma 7.1 it follows that the graph G is a q n ; q n ; q n=2 /-colored graph with q 1/ colors Theorem 2.10 and Theorem 2.11 now follow immediately from Theorem 2.4 and Theorem 2.7 Finally, it follows from Lemma 6.1 that X X X X IX B/ D IX B/ D o.1//jAjq; X 2A 2Fq 2Fq X2A which implies that there exists A0 A, A0 A, such that X IX B/ D o.1//q; 2Fq for any X A0 This completes the proof of Theorem 2.9 166 L A Vinh Product graphs – Proofs of Theorems 2.12, 2.14, and 2.16 For any non-degenerate bilinear form B ; / on Fqd , and for any element F , the product graph Bq;d / is defined as follows The vertex set of the product graph Bq;d / is the set V Bq;d // D F d n.0; : : : ; 0/: Two vertices a; b V Bq;d // are connected by an edge, a; b/ E.Bq;d //, if and only if B.a; b/ D When D 0, the graph is just a blow-up of a variant of Erd˝os–Rényi graph The eigenvalues of this graph are easy to compute (for example, see [1]) We will now study the product graph when F F , the product graph, Bq;d /, Lemma 8.1 For anypd and any element is a q d 1; q d ; 2q d /-graph Proof It is easy to see that Bq;d / is a regular graph of order q d and valency q d We now compute the eigenvalues of this multigraph (i.e graph with loops) For any a Ô b Fqd n.0; : : : ; 0/, the system B.a; x/ D ; x Fqd n.0; : : : ; 0/; B.b; x/ D ; has q d solutions when a Ô ˛b for all ˛ Fq , and no solution otherwise Hence, for any two vertices a Ô b, a and b have q d common neighbors if a and b are linearly independent, and no common neighbor otherwise Let A be the adjacency matrix of Pq;d / It follows that A2 D q d J C q d qd qd /I E; (8.1) where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph BE , where for any two vertices a; b V Bq;d //, a; b/ is an edge of BE if and only if a and b are linearly dependent Therefore, BE is a q 1/-regular graph, and all eigenvalues of E are at most q Since Bq;d / is a q d 1/-regular graph, q d is an eigenvalue of A with the all-one eigenvalue The graph Bq;d / is connected therefore the eigenvalue q d has multiplicity one For any a and b with B.a; b/ D , a and b are linearly independent, which implies that a and b have q d common neighbors Therefore, the graph is not bipartite and for any other eigenvalue  , jÂj < q d : Let v denote the corresponding eigenvector of  Note that v 1? , so J v D It follows from (8.1) that  qd C qd /v D q d Ev : 167 Sovability of equations over finite fields Hence, v is also an eigenvector of E Since all eigenvalues of E is bounded by q 1, we have Â Ä qd qd 1/q d C q < 2q d : The lemma follows Similarly as in the previous section, Theorems 2.12, 2.14, and 2.16 follow immediately from Corollary 3.2, Lemma 6.1, Theorem 2.7, and Lemma 8.1 Sum-product graphs For any non-degenerate bilinear form B ; / on Fqd and for any Fq , the sumproduct graph SBq;d / is defined as follows The vertex set of the sum-product graph SBq;d / is the set V SBq;d / D F F d : Two vertices U D a; b/ and V D c; d/ V SBq;d / are connected by an edge, U; V / E.SBq;d /, if and only if aCcC D B.b; d/ Our construction is similar to that of Solymosi in [29] Lemma 9.1 For and any p any d a q d C1 ; q d ; 2q d /-graph F , the sum-product graph, SBq;d /, is Proof It is easy to see that SBq;d / is a regular graph of order q d C1 and valency q d We now compute the eigenvalues of this multigraph For any a; c Fq and b Ô d Fqd , the system aCuC D B.b; v/; cCuC D B.d; v/; u Fq ; v Fqd ; has q d solutions (We can argue as follows There are q d possibilities of v such that B.b d; v/ D a c For each choice of v, there exists a unique u satisfying the system.) If b D d and a Ô c, then the system has no solution Hence, for any two vertices U D a; b/ and V D c; d/ V SBq;d //, if b Ô d, then U and V have exactly q d common neighbors, and if b D d and a Ô c, then U and V have no common neighbors Let A be the adjacency matrix of SBq;d / It follows that A2 D AAT D q d J C q d qd /I qd E; (9.1) where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph BE , where for any two vertices U D a; b/; V D c; d/ V SBq;d //; U; V / is an edge of BE if and only if we have a Ô c and b D d Since SBq;d / 168 L A Vinh is a q d -regular graph, it follows that q d is an eigenvalue of A with the all-one eigenvalue The graph SBq;d / is connected so the eigenvalue q d has multiplicity one Besides, choose b; d Fqd such that B.b; d/ D 2a Ô 0; then SBq;d / contains a triangle with three vertices a; 0/, a; b/, and a; d/, which implies that the graph is not bipartite Hence, for any other eigenvalue  , j j < q d Let v denote the corresponding eigenvector of  Note that v 1? , so J v D It follows from (9.1) that  qd C qd /v D q d Ev : Hence, v is also an eigenvector of E Since BE is a regular graph of order q the absolute value of any eigenvalue of E is at most q This implies that Â Ä qd qd C qd 1, 1/ < 2q d : q The lemma follows 9.1 Proofs of Theorems 2.19, 2.20 and 2.22 Similarly, Theorem 2.19 follows from Lemma 9.1 and Corollary 3.2; Theorem 2.22 follows from Theorem 2.4, Theorem 2.7 and Lemma 9.1 The proof of Theorem 2.20 remains Without loss of generality, we can suppose that … A For any A  Fq , let A D ¹1=a W a Aº Fq , EA D kA A A/d Fq Fqd , and FA D A/ 1=A/d Fq Fqd Let e.EA ; FA / be the number of edges of Gq;d between EA and FA It follows from Theorem 2.20 and Lemma 9.1 that q jEA jjFA j e.EA ; FA / Ä C 2q d jEA jjFA j: (9.2) q There is an edge between any two vertices a1 C Cad ; a1 b1 ; : : : ; ad bd / EA and ad ; b1 ; : : : ; bd 1 / FA Hence, e.EA ; FA / jAj2d : (9.3) Putting (9.2) and (9.3) together, we have jAj2d Ä Let x D jA Aj.d jAjd jA Ajd q 1/=2 jd Aj1=2 ; jd Aj C q q d jAjd jA Ajd jd Aj: then jAjd x C q d=2 x q jAj2d 0: Solving this inequality gives us the desired bound for x, concluding the proof of Theorem 2.20 Sovability of equations over finite fields 10 169 Finite Euclidean graphs – Proofs of Theorems 2.24 and 2.25 Let Q be a non-degenerate quadratic form on Fqd For any element Fq , the finite Euclidean graph Eq d; Q; / is defined as the graph with vertex set Fqd and the edge set E D ¹.x; y/ Fqd Fqd W x Ô y; Q.x y/ D : (10.1) p Recall that an n; d; /-graph is called a Ramanujan graph if Ä d p1 We also call an n; d; /-graph asymptotic Ramanujan graph if Ä C o.1// d when n; d; ! The spectrum of the finite Euclidean graphs Eq d; Q; / when Q.x/ D x12 C C xd2 was first investigated by Medrano et al [25], who proved that Eq d; Q; / are asymptotically Ramanujan for any Ô Bannai, Shimabukuro and Tanaka [6] extended this result to arbitrary non-degenerate quadratic form Q using the character tables of association schemes of affine type ([22]) The following theorem summaries the results from [6, Sections 2–6] and [22, Section 3] Theorem 10.1 ([6, 22]) Let Q be a non-degenerate quadratic form on Fqd For any Fq , the graph Eq d; Q; / is a q d ; C o.1//q d ; 2q d 1/=2 /-graph We are now ready to prove Theorems 2.24 and 2.25 We consider the graph G on the vertex set Fqd Two distinct vertices x; y are connected by an -colored edge if and only if Q.x y/ D (note that we only use q 1/ colors Fq ) From Theorem 10.1, the graph G is a q d ; C o.1//q d ; 2q d 1/=2 /-colored graph with q 1/ colors Theorems 2.24 and 2.25 follow immediately from Lemma 6.1 and Theorem 2.7 11 Finite non-Euclidean graphs – Proof of Theorem 2.26 Let Q be a non-degenerate quadratic form on Fqd For each element x S d Q/, we denote the pair of antipodes on S d Q/ containing x by Œx Let be the set of pairs of antipodes on the unit Q-sphere (or equivalently, the lines through them) For a fixed Fq , the finite non-Euclidean graph Pq;d / has the vertex set and the edge set ạ.x; y/ W x Ô y; Q.x y/ D ˙ º: We will see that our graphs are the same as the ones of Bannai, Hao, and Song in [5], and of Bannai, Shimabukuro, and Tanaka in [6] Note that can also be viewed as the set of all square-type non-isotropic onedimensional subspaces of Fqd with respect to the quadratic form Q The simple 170 L A Vinh orthogonal group Od Fq / acts transitively on and yields a symmetric association scheme ‰.Od Fq /; / of class q C 1/=2 We have two cases Case I Suppose that d D 2mC1 The relations of ‰.O2mC1 Fq /; / are given by for Ä i Ä q R1 D ¹.Œx; Œy/ W Q.x C y/ D 0º; Ri D ¹.Œx; Œy/ W Q.x C y/ D C i 1/ º 1/=2, and R.qC1/=2 D ¹.Œx; Œy/ W Q.x C y/ D 2º; where is a generator of the field Fq (see [5, Section 4]) Case II Suppose that d D 2m The relations of ‰.O2m Fq /; / are given by Ri D ¹.Œx; Œy/ for Ä i Ä q W Q.x C y/ D C i º 1/=2, and R.qC1/=2 D ¹.Œx; Œy/ W Q.x C y/ D 2º; where is a generator of the field Fq (see [5, Section 6] and [6, Section 2]) The graphs ; Ri / are not Ramanujan in general They, however, are asymptotic Ramanujan for large q The following theorem can be derived easily as in the proofs of [6, Theorem 2.2] and [6, Theorem 5.1] from the character tables of the association scheme ‰.Od Fq /; / ([5, Tables VI, VII] and [5, Theorem 6.3]) Note that [6, Theorem 2.2] requires an additional restriction that q D p r , where r is odd This restriction assures that the graphs ; Ri / Ä i Ä q C 1/=2) are Ramanujan if q is sufficiently large Since we only need our graphs to be asymptotic Ramanujan, we can apply [6, equation (3)] instead of [6, Lemma 2.1] in the proof of [6, Theorem 2.2] to remove this restriction Theorem 11.1 ([5, 6]) The graphs ; Ri / Ä i Ä q order q d 1 C o.1//=2 and valency q d C o.1// Let the graph ; Ri / with Ô valency of the graph Then j j Ä C o.1//q d 2/=2 1/=2/ are regular of be any eigenvalue of : Suppose that G is a graph with the vertex set , and the edge set is colored by ¹Ri º2Äi Ä.q 1/=2 Theorem 11.1 implies that G is a q d 1 C o.1//=2; q d C o.1//; 2Co.1//q d 2/=2 /-colored graph with q 3/=2 colors Theorem 2.26 now follows immediately from Theorem 2.7 171 Sovability of equations over finite fields 12 Further remarks In [10], Chapman et al proved results in Sections 2.3 and 2.5 using Fourier analytic methods Their proofs show that the conclusions of these results hold with the nondegenerate quadratic form Q is replaced by any function F with the property that the Fourier transform satisfies the decay estimates ˇ ˇ ˇ ˇˇ X ˇ ˇO ˇ ˇ d x m/ˇˇ C q d C1/=2 (12.1) ˇF t m/ˇ D ˇq x2Fqd WF x/Dt and ˇ ˇ ˇˇ ˇO ˇ ˇF t 0; : : : ; 0/ˇ D ˇˇq d X x2Fqd WF x/Dt ˇ ˇ x 0; : : : ; 0//ˇˇ C q ; (12.2) where s/ D e i Tr.s/=q m Ô 0; : : : ; 0/ Fqd and (recall that for y Fq , where q D p r with p prime, the trace of y is defined as r Tr.y/ D y C y p C C y p Fq ) The basic object in these proofs is the incidence function IB;C j / D jBjjC jv.j / D j.x; y/ B C W F x X D B.x/C.y/Fj x y/ D j j y/; x;y2Fqd where we denote by B, C , and Fj the characteristic function of the sets B, C and ¹x W F x/ D j º, respectively Using the Fourier inversion, we have IB;C j / D q 2d X O B.m/ CO m/FOj m/: (12.3) m2Fqd Now we define the F -distance graph GF q; d; j / with the vertex set V D Fqd and the edge set E D ạ.x; y/ V V W x Ô y; F x y/ D j º: Then the exponentials (or characters of the additive group Fqd )  à i Tr.x m/ em x/ D exp ; p (12.4) 172 L A Vinh for x; m Fqd , are eigenfunctions of the adjacency operator for the F -distance graph GF q; d; j / corresponding to the eigenvalue X em x/ D q d FOj m/: (12.5) m D F x/Dj Thus, the decay estimates (12.1) and (12.2) are equivalent to m C q d 1/=2 ; (12.6) (12.7) for m Ô 0; : : : ; 0/ Fqd , and 0;:::;0/ C qd : Let A denote the adjacency matrix of the graph GF q; d; j / with the orthonormal base v0 ; : : : ; vq d , corresponding to eigenvalues 0;:::;0/ ; : : : ; q 1;:::;q 1/ , p N n For any two sets B; C Fqd , let vB and vC be the characterwhere v0 D 1= P P istic vectors of B and C Let vB D i ˇi vi and vC D i i vi be their representations as linear combinations of v0 ; : : : ; vq d We have ÂX à ÂX à IB;C j / D eGF q;d;j / B; C / D vB AvC D ˇi vi A j vj i D ÂX j ˇi vi ÃÂX i D X à j j vj j i ˇi i : i From (12.3), (12.5) and the above expression, we can see the similarity between our approach and those in [10] as follows Given the decay estimates (12.1) and (12.2), we can bound the incidence function as X O IB;C j / jBjjC jFOj 0; : : : ; 0/ C q d 1/=2 q d jB.m/jj CO m/j mÔ.0;:::;0/ Cq jBjjC j !1=2 C C q d X 1/=2 d q !1=2 X O jB.m/j mÔ.0;:::;0/ jCO m/j2 mÔ.0;:::;0/ !1=2 Cq jBjjC j C C q d X jB.x/j x Cq jBjjC j C C q d p p jBj jC j: !1=2 X x jC.x/j 173 Sovability of equations over finite fields Given the bounds (12.6), (12.7) for eigenvalues of GF q; d; j /, we obtain the same bound for the incidence function q q X N N IB;C j / D 0;:::;0/ hvB ; 1= q d ihvC ; 1= qd i C m m m mÔ.0;:::;0/ Cq jBjjC j C C q d 1/=2 X jm jj mj mÔ.0;:::;0/ Cq jBjjC j C C q d D Cq jBjjC j C C q d 1/=2 kˇk2 k k2 p p 1/=2 jBj jC j: Our approach and the Fourier methods in [10] are basically both based on the above incidence bounds In a future paper, we will discuss the connection of these two methods in more detail Bibliography [1] N Alon and M Krivelevich, Constructive bounds for a Ramsey-type problem, Graphs Combin 13 (1997), 217–225 [2] N Alon and P Pudlák, Constructive lower bounds for off-diagonal Ramsey numbers, Israel J Math 122 (2001), 243–251 [3] N Alon, L Rònyai and T Szabò, Norm-graphs: Variations and applications, J Combin Theory Ser B 76 (1999), 280–290 [4] N Alon and J H Spencer, The Probabilistic Method, Second edition, Wiley-Interscience, Chichester, 2000 [5] E Bannai, S Hao and S.-Y Song, Character tables of the association schemes of finite orthogonal groups acting on the nonisotropic points, J Combin Theory Ser A 54 (1990), 164–200 [6] E Bannai, O Shimabukuro and H Tanaka, Finite analogues of non-Euclidean spaces and Ramanujan graphs, European J Combin 25 (2004), 243–259 [7] J Bourgain, A Szemeredi type theorem for sets of positive density, Israel J Math 54 (1986), no 3, 307–331 [8] J Bourgain, Mordell’s exponential sum estimate revisited, J Amer Math Soc 18 (2005), no 2, 477–499 [9] J Bourgain, N Katz and T Tao, A sum product estimate in finite fields and Applications, Geom Funct Anal 14 (2004), 27–57 [10] J Chapman, M B Erdogan, D Hart, A Iosevich and D Koh, Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates, Math Z., to appear 174 L A Vinh [11] D Covert, D Hart, A Iosevich and I Uriarte-Tuero, An analog of the FurstenbergKatznelson-Weiss theorem on triangles in sets of positive density in finite field geometries, Discrete Math., to appear [12] D Hart, A Iosevich, D Koh and M Rudnev, Averages over hyperplanes, sumproduct theory in vector spaces over finite fields and the Erd˝os-Falconer distance conjecture, Trans Amer Math Soc 363 (2011), 3255–3275 [13] H Furstenberg, Y Katznelson and B Weiss, Ergodic theory and configurations in sets of positive density, in: Mathematics of Ramsey Theory, Algorithms and Combinatorics 5, Springer-Verlag, Berlin (1990), 184–198 [14] M Z Garaev, The sum-product estimate for large subsets of prime fields, Proc Amer Math Soc 136 (2008), 2735–2739 [15] A A Glibichuk, Additive properties of product sets in an arbitrary finite fields, preprint [16] A A Glibichuk and S V Konyagin, Additive properties of product sets in fields of prime order, in: Additive Combinatorics, CRM Proceedings and Lecture Notes 43, American Mathematical Society, Providence, RI, (2007), 279–286 [17] K Gyarmati and A Sárközy, Equations in finite fields with restricted solution sets, II (algebraic equations), Acta Math Hungar 119 (2008), 259–280 [18] D Hart and A Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Anal Math 34 (2007) [19] A Iosevich and M Rudnev, Erd˝os distance problem in vector spaces over finite fields, Trans Amer Math Soc 359 (2007), 6127–6142 [20] H Iwaniec and E Kowalski, Analytic Number Theory, American Mathematical Society, Providence, RI, 2004 [21] M Krivelevich and B Sudakov, Pseudo-random graphs, in: More Sets, Graphs and Numbers, Bolyai Society Mathematical Studies 15, Springer-Verlag, Berlin, and János Bolyai Mathematical Society, Budapest (2006), 199–262 [22] W M Kwok, Character tables of association schemes of affine type, European J Combin 13 (1992), 167–185 [23] A Magyar, k-point configurations in sets of positive density of Zn , Duke Math J 146 (2009), no 1, 1–34 [24] A Magyar, On distance sets of large sets of integers points, Israel J Math 164 (2008), 251–263 [25] A Medrano, P Myers, H M Stark and A Terras, Finite analogues of Euclidean space, J Comput Appl Math 68 (1996), 221–238 [26] W G Schmidt, Equations over Finite Fields: An Elementary Approach, Lecture Notes in Mathematics 536, Springer-Verlag, Berlin, 1976 Sovability of equations over finite fields 175 [27] I Shparlinski, On the solvability of bilinear equations in finite fields, Glasg Math J 50 (2008), 523–529 [28] A Sárközy, On products and shifted products of residues modulo p, Integers (2008), no 2, Article A9 [29] J Solymosi, Incidences and the spectra of graphs, in: Building Bridges Between Mathematics and Computer Science, Bolyai Society Mathematical Studies 19, Springer-Verlag, Berlin (2008), 499–513 [30] L A Vinh, Explicit Ramsey graphs and Erd˝os distance problem over finite Euclidean and non-Euclidean spaces, Electron J Combin 15 (2008), no 1, Research Paper R5 [31] L A Vinh, On the sovability of systems of bilinear equations over finite fields, Proc Amer Math Soc 137 (2009), 2889–2898 [32] L A Vinh, A Szemerédi-Trotter type theorem and sum-product estimate over finite fields, European J Combin 32 (2011), no 8, 1177–1181 [33] L A Vinh, On the sovability of systems of sum-product equations over finite fields, Glasg Math J 53 (2011), no 3, 427–435 [34] L A Vinh, The Erd˝os-Falconer distance problem on the unit sphere in vector spaces over finite fields, SIAM J Discrete Math 25 (2011), no 2, 681–684 [35] L A Vinh, On kaleidoscopic pseudo-randomness of finite Euclidean graphs, Discuss Math Graph Theory, to appear [36] L A Vinh, Triangles in vector spaces over finite fields, Online J Anal Comb., to appear [37] V H Vu, Sum-product estimates via directed expanders, Math Res Lett 15 (2008), no 2, 375–388 Received October 1, 2011 Author information Le Anh Vinh, University of Education, Vietnam National University, Hanoi, Vietnam E-mail: vinhla@vnu.edu.vn Copyright of Forum Mathematicum is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use ... Ã2t d n copies of H The proof of Theorem 2.4 is similar to that of [21, Theorem 4.10], the proofs of Theorem 2.5 and Theorem 2.6 are similar to the proofs of Theorem 2.2 and Theorem 2.3, respectively... [31], the author studied the solvability of systems of bilinear equations over finite fields Following the proof of [21, Theorem 4.10], the author proved the following result 150 L A Vinh Theorem... another proof of this result using spectral graph methods In analogy with the study of simplices in vector spaces over finite fields, the author ([31]) studied the sovability of systems of bilinear