A class of quadratic matrix equations over finite fields
aa r X i v : . [ m a t h . R A ] S e p A CLASS OF QUADRATIC MATRIX EQUATIONS OVER FINITE FIELDS
YIN CHEN AND XINXIN ZHANGA
BSTRACT . We exhibit an explicit formula for the cardinality of solutions to a class of quadraticmatrix equations over finite fields. We prove that the orbits of these solutions under the naturalconjugation action of the general linear groups can be separated by classical conjugation invariantsdefined by characteristic polynomials. We also find a generating set for the vanishing ideal of theseorbits.
1. I
NTRODUCTION
Yang-Baxter matrix equations occupy a prominent place in pure mathematics and mathematicalphysics. Exploiting nontrivial solutions to a Yang-Baxter matrix equation over the complex fieldis a difficult task in general, whereas describing those solutions to some specific equations pre-cisely is indispensable in applications to algebraic geometry and statistical mechanics. Comparedto solving complex matrix equations, exploring solutions to a matrix equation over finite fields viaformulating an explicit formula for the cardinality of all solutions becomes realizable and has sub-stantial ramifications in the study of combinatorics and algebra, dating back to [Hod57], [Hod58]and [Hod64] etc. Our objectives of this article are to calculate the cardinality of solutions to aclass of matrix equations over finite fields, as well as to study the geometry of the orbits of thesesolutions under the natural conjugation action.Let F be a field and n ∈ N + be a positive integer. Given an n × n matrix A over F , the quadraticmatrix equation A · X · A = X · A · X called the parameter-independent Yang-Baxter equation over F , has been studied for the various cases where F is the field of complex numbers and A possesses some special properties; see for example [DD16], [DDH18] and the references therein.Throughout this article, F = F q denotes the finite field of order q = p s and we are interested insolving the parameter-independent Yang-Baxter equation over F q , when A = diag { a , . . . , a } is adiagonal matrix over F q .To articulate some extreme situations, we let M ( n , q ) denote the vector space of all n × n ma-trices over F q . If A is the zero matrix (i.e., a = X ∈ M ( n , q ) is a solution. Nowassume that a =
0. Since A commutes with every matrix in M ( n , q ) , we see that deciding whether X ∈ M ( n , q ) satisfies the parameter-independent Yang-Baxter equation is tantamount to verifyingwhether X is a solution of the following equation:( ∗ ) X − A · X = . We observe that the zero matrix and A itself are both solutions of this equation; in particular, if n =
1, the two solutions are all solutions as the left-hand side of ( ∗ ) is a polynomial in one variableof degree 2 in this case. Moreover, we also observe that for any n ∈ N + , if X is a nonsingularsolution, then X must be A . Denote by N ( n , q ) the set of all solutions to ( ∗ ) in M ( n , q ) . Thus | N ( n , q ) | − M ( n , q ) and the Date : September 16, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Matrix equations; general linear groups; finite fields; separating invariants. difficulty in determining | N ( n , q ) | is to find all nonzero singular n × n matrices satisfying theequation ( ∗ ).The bulk of the first two sections is to calculate the cardinality of those nonzero singular solu-tions to ( ∗ ). An elementary observation (Proposition 2.1) shows that N ( n , q ) could be endowedwith a conjugation action of the general linear group. This allows us to capitalize on the orbit-stabilizer formula and rational canonical forms of matrices to determine the number | N ( n , q ) | .After summarizing some preparations about classical conjugation invariants, rational canonicalforms, and computational steps, we close Section 2 with an explicit calculation for the case where n =
2; see Example 2.2. We will deal with the cases of higher dimensions ( n >
3) in Section 3. Toaccomplish this, the key is to reveal the concrete form of the rational canonical form of a nonzerosingular solution in N ( n , q ) ; see Lemma 3.2. As a consequence (Corollary 3.3), we prove, viaconstructing representatives in orbits, that the cardinality of the set O ( n , q ) of all orbits of N ( n , q ) under the conjugation action is equal to n +
1. Using the orbit-stabilizer formula, we finally derivean explicit formula on the cardinality | N ( n , q ) | ; see Theorem 3.6.In Section 4, we prove that the classical conjugation invariants ξ , . . . , ξ n separate the set O ( n , q ) of orbits (Theorem 4.3). Example 4.4 hints at the potential universality of our approach of sep-arating invariants in studying geometric properties of orbits. Consider the image points of theseorbits in F nq under the injection defined by ξ , . . . , ξ n . We find an ideal I n of F q [ x , . . . , x n ] , viagiving explicit generators, such that the variety of I n in F nq coincides with the image of O ( n , q ) ;see Theorem 4.7. A surprising result appears in Proposition 4.6, showing that the ideal I n couldbe generated by (cid:0) n + (cid:1) quadratic polynomials. Conventions.
Throughout this article, N + denotes the set of all positive integers. Let I n be theidentity matrix of rank n ∈ N + . For B ∈ M ( k , q ) and C ∈ M ( ℓ, q ) , we use B ⊕ C to denote theblock matrix (cid:0) B C (cid:1) in M ( k + ℓ, q ) . Acknowledgements.
This research was partially supported by NNSF of China (No. 11401087).2. C
ONJUGATION A CTIONS AND R ATIONAL C ANONICAL F ORMS
In this preliminary section, we let n > ( n , q ) be the general linear group of degree n over F q . Recall that the conjugation action of GL ( n , q ) on M ( n , q ) is defined by ( P , X ) P · X · P − for P ∈ GL ( n , q ) and X ∈ M ( n , q ) . We write [ X ] for the conjugacy class of X . Moreover, thecharacteristic polynomial of X ∈ M ( n , q ) is defined as(2.1) det ( λ · I n − X ) = λ n + n ∑ i = ( − ) i · ξ i ( X ) · λ n − i where λ is an indeterminate and the coefficients ξ , ξ , . . . , ξ n are algebraically independent invari-ants in the invariant ring F q [ M ( n , q )] GL ( n , q ) . In particular, ξ and ξ n are just the well-known traceand determinant functions respectively. Note that unlike the classical case (over the complex field),these ξ i here do not generate the invariant ring; see [Smi02, Theorem 1.1] for the case n = ( n , q ) on M ( n , q ) restricts to anaction on N ( n , q ) . We denote by O ( n , q ) the set of orbits of N ( n , q ) under this action. Proposition 2.1.
If an n × n matrix X ∈ N ( n , q ) , then Y ∈ N ( n , q ) for all Y ∈ [ X ] .Proof. Suppose that Y = P · X · P − for some P ∈ GL ( n , q ) . Since X = A · X , we see that Y − A · Y = ( P · X · P − ) − A · P · X · P − = P · X · P − − P · A · X · P − = P · ( X − A · X ) · P − =
0. Hence, Y ∈ N ( n , q ) . (cid:3) CLASS OF QUADRATIC MATRIX EQUATIONS OVER FINITE FIELDS 3
Consider a monic polynomial f ( x ) = x k + ∑ k − i = a i · x i ∈ F q [ x ] . The companion matrix of f ( x ) isdefined as(2.2) C ( f ) : = · · ·
00 0 1 . . . ...... . . . . . . . . . 00 0 · · · − a − a − a · · · − a k − for k > C ( f ) : = ( − a ) for k =
1. Recall that every matrix X ∈ M ( n , q ) is similar to a diagonalblock matrix of the form C ( f ) ⊕ C ( f ) ⊕ · · · ⊕ C ( f r ) , called the rational canonical form of X ,where f ( x ) , . . . , f r ( x ) ∈ F q [ x ] are monic polynomials and f i ( x ) divides f i + ( x ) for i = , , . . . , r − X is in N ( n , q ) ,we may assume that X = C ( f ) ⊕ C ( f ) ⊕ · · · ⊕ C ( f r ) and further, we write A = A ⊕ A ⊕ · · · ⊕ A r as a block matrix such that the sizes of A i and C ( f i ) are same for each i . Clearly, ( ∗ ) is completelydetermined by the system of equations:(2.3) C ( f i ) − A i · C ( f i ) = i = , , . . . , r .Based on these observations, we may proceed the following steps to determine the cardinality | N ( n , q ) | , i.e., the number of solutions to ( ∗ ).(1) Determine all possible nonzero singular rational canonical forms X , . . . , X t of n × n matri-ces.(2) Find those X j from { X , . . . , X t } for which the system (2.3) of equations follows, and denoteby X , . . . , X ℓ (relabelling if necessary), where ℓ = | O ( n , q ) | − ℓ t .(3) For i ∈ { , . . . , ℓ } , calculate the order of the stabilizer subgroup GL ( n , q ) X i of X i in GL ( n , q ) .Since the number of all nonzero singular solutions to ( ∗ ) equals ∑ ℓ i = | [ X i ] | and | GL ( n , q ) | = | [ X i ] | · | GL ( n , q ) X i | , it follows that(2.4) | N ( n , q ) | = + ℓ ∑ i = | [ X i ] | = + ℓ ∑ i = | GL ( n , q ) || GL ( n , q ) X i | . We conclude this section with the following example that not only illustrates the above procedurebut also serves to higher dimension cases in Section 3.
Example 2.2 ( n = . There are two possible rational canonical forms: (cid:16) − a − a (cid:17) and (cid:0) − a − a (cid:1) for a , a ∈ F q . As the first canonical form is either zero or nonsingular, the second one is theunique canonical form for nonzero singular solutions. Note that its determinant is a , thus a = X = (cid:0) − a (cid:1) is an arbitrary nonzero singular solution. Substitut-ing C ( f i ) in (2.3) with X , we have0 = (cid:18) − a (cid:19) − (cid:18) a a (cid:19) (cid:18) − a (cid:19) = (cid:18) − a − a a + aa (cid:19) which implies that X = (cid:0) a (cid:1) . To determine | [ X ] | , we need to determine the order of the stabilizersubgroup GL ( , q ) X . Here we take a direct approach to do that. Let P = (cid:0) e bd c (cid:1) ∈ GL ( , q ) X beany element. As P · X · P − = X , it follows that0 = (cid:18) e bd c (cid:19) (cid:18) a (cid:19) − (cid:18) a (cid:19) (cid:18) e bd c (cid:19) = (cid:18) − d e + ab − c − ad d (cid:19) . YIN CHEN AND XINXIN ZHANG
Thus P = (cid:0) c − ab b c (cid:1) . Since P is invertible, we see that c = b = c / a . Hence, | GL ( , q ) X | =( q − ) . Recall that | GL ( , q ) | = ( q − )( q − q ) . Therefore | [ X ] | = | GL ( , q ) || GL ( , q ) X | = ( q − )( q − q )( q − ) = q + q and | N ( , q ) | = q + q + . ✸ | N ( n , q ) | ( n > ) In this section, we will determine the cardinality of N ( n , q ) . Let n > X ∈ M ( n , q ) be amatrix. Usually, it is difficult to determine the rational canonical form for X precisely. However,with the assumption that X ∈ N ( n , q ) , the following lemma shows that the canonical form of X will be built by rational canonical blocks of size less than or equal to 2. Lemma 3.1.
Let f ( x ) ∈ F q [ x ] be a monic polynomial of degree k > . Then C ( f ) = A · C ( f ) .Proof. A direct calculation shows that the entry at the first row and third column in C ( f ) will be1. However, the entry at the same position in A · C ( f ) is zero. Hence, C ( f ) and A · C ( f ) are neverequal. (cid:3) Note that we have determined nonzero singular rational canonical blocks of size 2 satisfying(2.3) in Example 2.2. Throughout this section, we let Q ( a ) : = (cid:0) a (cid:1) and P m ( b ) be the diagonalmatrix of size m with the diagonals b for m n and b ∈ F q [ λ ] , where λ is an indeterminate. For k ∈ { , . . . , ⌊ n / ⌋} , we define Q k ( a ) to be the direct sum of k copies of Q ( a ) and let X k ( b , a ) : = P n − k ( b ) ⊕ Q k ( a ) . Note that when n = m is even, we make the convention that X m ( b , a ) = Q m ( a ) for any b . Lemma 3.2.
Let X ∈ N ( n , q ) be a nonzero singular matrix. Then X is similar to either X k ( , a ) orX k ( a , a ) for some k ∈ { , . . . , ⌊ n / ⌋} .Proof. We use X to denote the rational canonical form of X . By Lemma 3.1, the blocks ap-peared in X are of size either 1 or 2. If these blocks are all 1 ×
1, we may assume that X = diag {− a , − a , . . . , − a n } . Note that x + a i divides x + a i + for i = , . . . , n −
1, thus a = · · · = a n .Hence, X is either zero or nonsingular, contradicting with that X is nonzero singular. This meansthat X contains at least one block of size 2. Now we suppose that X = diag {− a , − a , . . . , − a n − k } M (cid:16) ⊕ ki = (cid:16) − a i , − a i , (cid:17)(cid:17) for some k ∈ { , . . . , ⌊ n / ⌋} . As before, since x + a i divides x + a i + for i = , . . . , n − k −
1, itfollows that a = · · · = a n − k . By (2.3), we see that X = P n − k ( b ) M (cid:16) ⊕ ki = (cid:16) − a i , − a i , (cid:17)(cid:17) where b = a . Let f i = x + a i , x + a i , be the polynomial with C ( f i ) = (cid:16) − a i , − a i , (cid:17) . Since f i + is divisible by f i for each i = , . . . , k −
1, we see that f = f = · · · = f k . Thus X = P n − k ( b ) M (cid:16) ⊕ ki = (cid:0) − a − a (cid:1)(cid:17) for some a , a ∈ F q . Note that the polynomial corresponding the ( n − k ) -th block of X is x andthe polynomial corresponding the ( n − k + ) -th block is x + a x + a . Being divisible by x for CLASS OF QUADRATIC MATRIX EQUATIONS OVER FINITE FIELDS 5 x + a x + a implies that a =
0. Applying (2.3) again, it follows from Example 2.2 that a = − a .Therefore, X = X k ( b , a ) , where b = a . (cid:3) Corollary 3.3.
The cardinality of O ( n , q ) is n + .Proof. If n = m is even, then O ( n , q ) = { [ P n ( )] , [ P n ( a )] , [ Q m ( a )] , [ X k ( , a )] , [ X k ( a , a )] | k m − } . Thus | O ( n , q ) | = ( m − ) + = n +
1. If n = m + O ( n , q ) = { [ P n ( )] , [ P n ( a )] , [ X k ( , a )] , [ X k ( a , a )] | k m } . Thus | O ( n , q ) | = m + = n + (cid:3) Lemma 3.4.
If k ∈ { , . . . , ⌊ n / ⌋} , then (1) the elementary divisors of X k ( , a ) consist of n − k copies of λ and k copies of λ − a; and (2) the elementary divisors of X k ( a , a ) consist of n − k copies of λ − a and k copies of λ .Proof. Here all λ -matrices involved will be working over the polynomial ring F q [ λ ] . Note thatas a is invertible, the λ -matrix (cid:16) λ − λ − a (cid:17) of Q ( a ) could be diagonalized via applying elementarytransformations. In fact, (cid:16) − a (cid:17) (cid:16) λ − λ − a (cid:17) (cid:16) a (cid:17) = (cid:0) λ λ − a (cid:1) = : e Q ( a ) . Let e Q k ( a ) denote the directsum of k copies of e Q ( a ) .(1) We first capitalize on [Bro93, Lemma 16.12] to find all invariant factors of X k ( , a ) . Clearly,the λ -matrix λ · I n − X k ( , a ) is equivalent to P n − k ( λ ) ⊕ e Q k ( a ) , which has the Smith normal formdiag { h ( λ ) , h ( λ ) , . . . , h n ( λ ) } , we say. Here h i ( λ ) divides h i + ( λ ) for i = , , . . . , n −
1. Sincethe values of minors of order i of P n − k ( λ ) ⊕ e Q k ( a ) are of forms λ j ( λ − a ) i − j with 0 j i , itfollows that h ( λ ) = · · · = h k ( λ ) = h n ( λ ) = λ n − k ( λ − a ) k . For k + i n −
1, we have h i ( λ ) = ( λ i − k , i n − k , λ i − k ( λ − a ) i − ( n − k ) , i > n − k . Hence, the invariant factors of λ · I n − X k ( , a ) consist of { , . . . , | {z } k , λ , . . . , λ | {z } n − k , λ ( λ − a ) , . . . , λ ( λ − a ) | {z } k } and the elementary factors contains n − k copies of λ and k copies of λ − a .(2) Similarly, we note that λ · I n − X k ( a , a ) is equivalent to P n − k ( λ − a ) ⊕ e Q k ( a ) and assumethat the corresponding Smith normal form is diag { h ( λ ) , h ( λ ) , . . . , h n ( λ ) } . Observe that h ( λ ) = · · · = h k ( λ ) = h n ( λ ) = λ k ( λ − a ) n − k . Switching the roles of λ and λ − a in the previouscase, we see that for k + i n − h i ( λ ) = ( ( λ − a ) i − k , i n − k , ( λ − a ) i − k λ i − ( n − k ) , i > n − k . Hence, the corresponding elementary factors consist of n − k copies of λ − a and k copies of λ . (cid:3) Corollary 3.5.
For each k ∈ { , . . . , ⌊ n / ⌋} and b ∈ { , a } , we have | GL ( n , q ) X k ( b , a ) | = | GL ( n − k , q ) | · | GL ( k , q ) | . Proof.
It follows from Lemma 3.4 and [Hou18, Theorem 6.14] that | GL ( n , q ) X k ( b , a ) | = q ( n − k ) + k · n − k ∏ i = ( − q − i ) · k ∏ j = ( − q − j ) YIN CHEN AND XINXIN ZHANG which is exactly equal to the product of the orders of GL ( n − k , q ) and GL ( k , q ) . (cid:3) Together (2.4), Corollary 3.5 and Example 2.2 immediately imply that
Theorem 3.6.
For n > , we have | N ( n , q ) | = + | GL ( m , q ) | · (cid:16) | GL ( m , q ) | + ∑ m − k = | GL ( m − k , q ) |·| GL ( k , q ) | (cid:17) , n = m , | GL ( m + , q ) | · ∑ mk = | GL ( m − k + , q ) |·| GL ( k , q ) | , n = m + , where | GL ( ℓ, q ) | = ∏ ℓ − i = ( q ℓ − q i ) for every ℓ ∈ N + . Note that here the construction of orbits in Corollary 3.3 has been applied. We conclude thissection by showcasing | N ( n , q ) | for several small n . Example 3.7. (1) | N ( , q ) | = q + q + | N ( , q ) | = q ( q + q + ) + | N ( , q ) | = q ( q + )( q + q + q + ) + | N ( , q ) | = q ( q − q + )( q + q + )( q + q + q + q + ) + ✸
4. S
EPARATING I NVARIANTS
In this section, we separate the orbits via invariants and find a generating set for the vanishingideal of these orbits. Consider the set O ( n , q ) of orbits and the classical conjugation invariants ξ , ξ , . . . , ξ n . The map ξ : O ( n , q ) −→ F nq given by X ( ξ ( X ) , ξ ( X ) , . . . , ξ n ( X )) is well-defined.Let A be the set of all functions from O ( n , q ) to F q . We say that a subset B ⊆ A is separating for O ( n , q ) if for any two distinct orbits X , Y ∈ O ( n , q ) , there exists a function f ∈ B such that f ( X ) = f ( Y ) ; see [DK15, Section 2.4] for more details on separating invariants. Lemma 4.1.
The map ξ is injective if and only if { ξ , ξ , . . . , ξ n } is separating for O ( n , q ) .Proof. Assume that ξ is injective and { ξ , ξ , . . . , ξ n } is not separating. Then there exist two dis-tinct orbits X , Y ∈ O ( n , q ) such that ξ i ( X ) = ξ i ( Y ) for all i = , . . . , n . Thus ξ ( X ) = ξ ( Y ) , whichcontradicts with the assumption that ξ is injective. Conversely, if { ξ , ξ , . . . , ξ n } is separating, thenfor any two distinct orbits X , Y ∈ O ( n , q ) , there exists some i ∈ { , . . . , n } such that ξ i ( X ) = ξ i ( Y ) .Thus ξ ( X ) = ξ ( Y ) and ξ is injective. (cid:3) Lemma 4.2.
The cardinality of the image of ξ is equal to n + .Proof. Let ϕ λ ( X ) : = det ( λ · I n − X ) be the characteristic polynomial of a matrix X ∈ M ( n , q ) . Notethat ϕ λ ( P n ( )) = λ n and ϕ λ ( P n ( a )) = ( λ − a ) n . Thus ξ ([ P n ( )]) = ( , , . . . , ) and ξ ([ P n ( a )]) = (cid:0) n · a , (cid:0) n (cid:1) · a , . . . , a n (cid:1) . By Corollary 3.3, for each remaining orbit [ X ] ∈ O ( n , q ) \ { [ P n ( )] , [ P n ( a )] } ,there exist some k ∈ { , . . . , ⌊ n / ⌋} such that ϕ λ ( X ) = ( λ − b ) n − k · λ k · ( λ − a ) k where b ∈ { , a } .Assume that n = m is even. For 1 k m −
1, we have ϕ λ ( X k ( , a )) = λ m − k · ( λ − a ) k and ξ ([ X k ( , a )]) = (cid:16) k · a , (cid:0) k (cid:1) · a , . . . , a k , , . . . , (cid:17) . Since ϕ λ ( Q m ( a )) = λ m · ( λ − a ) m , it follows that ξ ([ Q m ( a )]) = (cid:18) m · a , (cid:18) m (cid:19) · a , . . . , a m , , . . . , (cid:19) . Furthermore, as ϕ λ ( X k ( a , a )) = λ k · ( λ − a ) m − k , we see that ξ ([ X k ( a , a )]) = (cid:18) ( m − k ) · a , (cid:18) m − k (cid:19) · a , . . . , a m − k , , . . . , (cid:19) CLASS OF QUADRATIC MATRIX EQUATIONS OVER FINITE FIELDS 7 for 1 k m −
1. Consider the ordered sequence ξ ([ P m ( )]) , ξ ([ X ( , a )]) , . . ., ξ ([ X m − ( , a )]) , ξ ([ Q m ( a )]) , ξ ([ X m − ( a , a )]) , . . ., ξ ([ X ( a , a )]) , ξ ([ P m ( a )]) . Arraying the last 2 m items into rows,we obtain a 2 m × m lower triangular matrix: a · · · ∗ a . . . ...... . . . . . . 0 ∗ · · · · · · a m which is invertible as a = . This fact shows that the map ξ evaluating on O ( m , q ) has 2 m + n = m + ξ is equal to n + (cid:3) Theorem 4.3.
The set { ξ , ξ , . . . , ξ n } is separating for O ( n , q ) . Moreover, if p > n, then ξ canseparate orbits in O ( n , q ) .Proof. Lemma 4.2 together with Corollary 3.3 implies that ξ is injective. By Lemma 4.1, we seethat ξ , ξ , . . . , ξ n separate the orbit set O ( n , q ) . For the second statement, we note in the proof ofLemma 4.2 that { ξ ( X ) | X ∈ O ( n , q ) } = { k · a | k n } , which has cardinality n +
1, by theassumption p > n . Hence, ξ separates all orbits in O ( n , q ) . (cid:3) The following example illustrates that when p n , ξ , . . . , ξ n might not be superfluous. Example 4.4.
Suppose that n =
3. Then O ( , q ) = { [ P ( )] , [ X ( , a )] , [ X ( a , a )] , [ P ( a )] } and thevalues of ξ on elements of O ( , q ) are: ( , , ) , ( a , , ) , ( a , a , ) , ( a , a , a ) respectively.(1) If p =
2, then either ξ or ξ can not separate the orbits [ P ( )] and [ X ( a , a )] . Hence, ξ cannot be removed in this case. However, ξ is superfluous. In fact, the map O ( , q ) −→ F q , X ( ξ ( X ) , ξ ( X )) is injective. Via this injection, we observe that orbits of O ( , q ) forms a rectangle in the plane F q : ( , )( , a ) ( a , )( a , a ) where the four points ( , ) , ( a , ) , ( , a ) , ( a , a ) correspond to [ P ( )] , [ X ( , a )] , [ X ( a , a )] , [ P ( a )] in O ( , q ) respectively.(2) Assume that p =
3. The functions ξ and ξ can not separate the orbits [ P ( )] and [ P ( a )] .Thus ξ is necessary in this case. After removing ξ in ξ , the injective map O ( , q ) −→ F q definedby X ( ξ ( X ) , ξ ( X )) embeds O ( , q ) into an isosceles triangle in F q : ( , )( , a ) ( a , )( − a , ) YIN CHEN AND XINXIN ZHANG where ( , ) , ( a , ) , ( − a , ) , ( , a ) correspond to [ P ( )] , [ X ( , a )] , [ X ( a , a )] , [ P ( a )] in O ( , q ) re-spectively. ✸ We look back at the image points of O ( n , q ) in F nq via the map ξ . As in the proof of Lemma 4.2,we use v , v , . . . , v n to denote these points respectively. More precisely, v = ( , , . . . , ) and v = ( a , , . . . , ) , v = (cid:18)(cid:18) (cid:19) a , (cid:18) (cid:19) a , , . . . , (cid:19) , . . . , v n = (cid:18)(cid:18) n (cid:19) a , (cid:18) n (cid:19) a , . . . , (cid:18) nn (cid:19) a n (cid:19) . The rest of this section is devoted to finding an ideal I n ⊆ A n : = F q [ x , . . . , x n ] such that V n : = { v i | i = , , . . . , n } is the variety (i.e., set of zeros) of I n in F nq . Throughout we denote by V ( I n ) thevariety of I n .We start with the case n = Proposition 4.5.
Let I be the ideal of A generated by B : = { f : = x − a x , f : = x x − ax , f : = x − ax − x } . Then V ( I ) = V .Proof. Assume that v = ( c , c ) ∈ V ( I ) is any element. Since f ( v ) =
0, we see that c is equalto either 0 or a . If c =
0, then f ( v ) = c , and the fact that f ( v ) = c ∈ { , a } . If c = a , it follows from the fact that f ( v ) = c = a . Clearly, the valuationof f at ( a , a ) is zero. Hence, V ( I ) = { ( , ) , ( a , ) , ( a , a ) } = V . (cid:3) We regard A ⊆ · · · ⊆ A k ⊆ A k + ⊆ · · · ⊆ A n as a sequence of containments of F q -subalgebrasof A n . For n >
3, we define B n : = (cid:26) f − f ( w n ) a n · x n | f ∈ B n − (cid:27) ∪ (cid:26) x n · x i − (cid:18) ni (cid:19) a i · x n | i = , . . . , n (cid:27) where w n : = (cid:0)(cid:0) n (cid:1) a , (cid:0) n (cid:1) a , . . . , (cid:0) nn − (cid:1) a n − (cid:1) ∈ F n − q . Proposition 4.6.
For each n > , we have | B n | = (cid:0) n + (cid:1) .Proof. We may assume that n > n = f ∈ B n − does not involve x n . By the definition of B n , we see that | B n | = | B n − | + n . Since theinduction hypothesis implies that | B n − | = (cid:0) n (cid:1) , it follows that | B n | = (cid:0) n (cid:1) + (cid:0) n (cid:1) = (cid:0) n + (cid:1) . (cid:3) For example, when n =
3, we see that w = ( a , a ) , | B | = B = { f , f − x , f − a · x } ∪ { x − a · x , x x − a · x , x x − a · x } . Theorem 4.7.
Let n > and I n be the ideal of A n generated by B n . Then V ( I n ) = V n .Proof. We may assume that n >
3. Given a vector v ∈ F nq , we denote by e v the projection image of v onto F n − q via removing the last component of v . We first show that each v i ∈ V n belongs to V ( I n ) .Indeed, for 1 i n , we see that the valuation ( x n · x i − (cid:0) ni (cid:1) a i · x n ) | v n = a n · (cid:0) ni (cid:1) a i − (cid:0) ni (cid:1) a i · a n = . Further, for f ∈ B n − , note that f does not involve x n and e v n = w n , thus ( f − f ( w n ) a n · x n ) | v n = f ( v n ) − f ( w n ) = f ( e v n ) − f ( w n ) = . This shows that v n ∈ V ( I n ) . Moreover, since the last components of v , v , . . . , v n − are are zero and the induction hypothesis implies that { e v i | i = , , . . . , n − } ⊆ V ( I n − ) , we deduce that the valuation of each f ∈ B n at v i is equal to zero for i ∈ { , , . . . , n − } .This proves that V n ⊆ V ( I n ) .Conversely, since | V n | = n +
1, it suffices to show that | V ( I n ) | = n +
1. Suppose v = ( c , c , . . . , c n ) ∈ V ( I n ) denotes an arbitrary element. Since x n − a n · x n ∈ B n , it follows that c n − a n · c n =
0, which
CLASS OF QUADRATIC MATRIX EQUATIONS OVER FINITE FIELDS 9 implies that c n must be in { , a n } . If c n = a n , then v = v n ∈ V ( I n ) is unique; and assume that c n = v ∈ V ( I n ) if and only if e v ∈ V ( I n − ) . Thus | V ( I n ) | = | V ( I n − ) | + = n +
1, as desired. Herethe last equation holds from the induction hypothesis that | V ( I n − ) | = n . (cid:3) R EFERENCES [Bro93] William C. Brown,
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Invariants of × -matrices over finite fields , Finite Fields Appl. (2002), no. 4, 504–510.S CHOOL OF M ATHEMATICS AND S TATISTICS , N
ORTHEAST N ORMAL U NIVERSITY , C
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