Necessary Conditions for Hyperbolic Systems: A Rigorous Analysis

We prove necessary conditions for the well posedness of the Cauchy problem for a general class of hyperbolic systems with multiple characteristics. No assumption on the rank of the principal symbol on the multiple characteristic set is made. Basically our conditions are Ivrii-Petkov type vanishing conditions for the symbol of a suitably defined non commutative determinant of the full symbol of the system 1. Introduction In this note we study the Cauchy problem for a first order system n defined in a neighborhood of the origin of W + 1 with local coordinates c c ) where Aj@) and B@) are m x m smooth matrices and n Our aim is to obtain general necessary conditions at multiple characteristic points in order that the Cauchy problem for L@, D) is coo well posed. Let p be a multiple characteristic point. Then possibly using the standard block diagonalization procedure, we may assume that p is a characteristic point of multiplicity m. Without any generality restrictions we may assume that p = (0, en) where 0, 1) e W + 1 . In order to obtain necessary conditions we make a dilation around the reference characteristic point p. This procedure localizes the operator at that point. The symbol of the dilated (localized) operator can be thought of as a formal expansion in the dilation parameter and winds up in some non commutative field. We then introduce a determinant "Det(s) " , which depends on s e N, on this non commutative field (the precise definition will be given in BovE Section 2). A general outline of our result can be sketched as follows: the leading part of the non commutative determinant "Det(s)" of the dilation (localization) of the complete symbol should be the dilation (localization) of the usual determinant of the principal symbol. In our previous paper [4] we proved this result under the assumption that the rank of LI(O, en) is m — 2. In this note we remove the restriction on the rank. In Section 2 we define Det(s) for matrix valued symbols j=n(A) Det(s) A has the form j=p(A) where the fj@, e) are holomorphic at (0, 0)• Moreover Det(s) verifies, for instance, the following relation: let f; A) and B@, e; X) be polynomials in @, f) and let A-SD; A) = A(c, A -SD; v s D; A) then Det(s) C = Det(s) A • Det(s) B. We note that det en = —> oo (1.1) hp(c, G) = la+ßl=rn THEOREM 1.1. Let 2s + 2 2 m, s e N and put F(c, e; A) = Ll(F 1 c, + A -1 () + 0(Ä-m ), F) = hp (1.2) where c(A) denotes the leading part of the A-ecpansion of A. The method of proof of the above theorem consists in constructing an asymptotic solution ux of the equation where q verifies 1 — q — ö — s — ö, that is q = s + 2, which contradicts an a priori estimate (depending on A) resulting from the coo well posedness of the Cauchy problem for Ll@, D) + Lo@). Here is an outline of the paper: in Section 2 we give the precise definition of Det(s) and state several properties proved in [4] which will be used in later sections. In Section 3, we prove that one can construct an asymptotic solution for a m x m system F@, X) provided det F@, e; A) = C) with g@, 0) = 0, 0) # 0 and k < s (Proposition 3.1). We prove Theorem 1.1 in Section 4 applying Proposition 3.1. In we give some remarks on Det(s) and give some concrete examples. Finally the authors would like to express their sincere gratitude to the referee for his careful reading and the remarks, which helped to improve the text. 2. Definition of Det(s) We denote by the ring of all formal Laurent series in with coeffcients which are meromorphic in a neighborhood of (0, 0). We define for f, g e K (X) by -k where €) = D? Dßcf(c, e), Dc3 Bovn In particular, if f@, e; X), g@, G; A) e K (A) are given by a finite sum of fj@, €) and gj@, e) respectively and both fj@, f) and gj@, () are polynomials in f, then it is clear that A -S D; A -S D; X) = A-S D; A). It is easy to see that K (A) is a non commutative field when equipped with the product Let us set where [K (A) x K(X) X ] denotes the commutator subgroup of the multiplicative group = K (A) \ {0}. Recall that every f # 0 has a canonical image f* in K*. Let us denote by M (m; K (X)) the set of all m x m matrices with entries in K (X). Dieudonné ( [61) (see [1] for an exposition of the theory) proved that there exists a unique multiplicative morphism verifying the following properties: Let A e M (m; K(Å)) and let A/ be obtained from A by multiplying one row by f G K (A) then Let N be obtained from A by adding one row to another then A' = Deqs) A. 3. The notation Detts) stresses the fact that the non commutative determinant depends on s. Before defining Det(s) we recall Lemma 2.1 and Lemma 2.2 in [4]. LEMMA 2.1. Let f, g G K(A). Assume 1 + 00 —s) then we have fg = 1 + and vice versa. LEMMA 2.2. Let f = g Ej-ng e K(A) and assume that —1 — 1, e K ( X) verifies 1 + 0(Ä-S)• Then we have fi = gi, rif S i < nf + s. (2.1) Let f¯l ,g ¯l e K(X) be such that —1 = 1. Then it is clear that = fg + oonf-+-ng—s —1 and this shows that From Lemma 2.1 has the form Taking these facts into account we set R - { I is meromorphic in a neighborhood of (O, O) and define a morphism 7T : K* —+ R C K(X) by j=nj where f = E j=nf fj@, f) is a representative of f*. Since Lemma 2.2 shows that e [K (A) x K (A) X ] implies (2.1), T is well defined. It is also easy to see that is a morphism when K is equipped with the product for = .nf+s—l G) e R. We now define / dj=nj Det(s) A = T(Detts A) so that Det(s) verifies all properties listed as (1), (2) and (3) above. We also note that = Det(s) A • Det(s) B (2.2) and Det(s) A = Det(s) All • Det(s) A22 (2.3) Al 1 A 12 All if A = A21 2422 In what follows we write Det whenever the role and the value of the parameter s are evident from the context. DEFINITION 2.1. Let q = s + 2. We say that A e M (m; K (x)) belongs to H(k) if A = E and Aj is a sum of terms Ajz which are homogeneous of degree j + k — Eq 2 0 for some t e N. We define ö(A) for A e 'H(k) by that is, ö(A) is the sum of the terms with the highest degree of homogeneity. It is easy to check that if Ai e H(ki) then A1#A2 G H(kl + k2). We just state without proof a number of results that are used extensively below. For a proof we refer to the paper [4]. LEMMA 2.3. Let Ai e H(ki), • = 15 2. Then we have LEMMA 2.4. Assume that G = Then we have DetG = det G + o(F S ). In particular Det G = 0(1). Let f be holomorphic at (0, 0). We say that F e K (X), F = E is holomorphic outside {f = 0} if for every j, writing Fj = fj/gj, where fj and gj are relatively prime, the irreducible factors of gj consist of those of f so that fnj /gj is holomorphic at (0, 0) for some nj e N. We say that F is holomorphic on {f = 0} if, for every j, gj and f have no common irreducible factor. The same definitions are available for F e R because K C K (X). Let F = (Fij) e M (m; K (A)). Then we say that F is holomorphic outside {f = 0} if every Fij is holomorphic outside {f = 0} and we say that F is holomorphic on {f = 0} if every Fij is holomorphic on {f = 0}. PROPOSITION 2.1. If F is holomorphic outside {f = 0} then Det F is holomorphic outside {f 0}. COROLLARY 2.1. If F is holomorphic at (0, 0) then DetF is also holomorphic at (0, 0). 3. A lemma In this section we are interested in constructing an asymptotic null solution for an operator F@, F S D; A) whose symbol has the form where Fj@, C) is a m x m matrix which is a polynomial in and real analytic in in some open set U. Our purpose is to prove the following PROPOSITION 3.1. Assume that for e U we have for some k e N, 0 k < s, where g@, 0) 0 and Deog@, 0) # 0. Then one can find an open set V C U and smooth functions defined in V such that +j(jo, c') = 0 for a certain with (jo, c') e V and that the rank of FOG, 0) is if whenever j > every j x j minor of Fo@, 0) has an identically vanishing determinant and there is a x minor whose determinant is not identically zero. Assume that rank Fo(c, 0) = m — r, 1 r m. Then there are non singular real analytic matrices M@), N@) defined in some open set contained in U such that o where 1m—r is the (m — r) x (m — r) identity matrix. Writing A -S D; A) = A-SD; det F (x, e; A) = with c@) = det M@)N@) # 0 since k < s. Obviously it is enough to construct an asymptotic null solution for F (c, A). Denote F(c, C; X), e) by F@, e; X) and g@, f) again. Let us set Fll F12 F21 F22o where the block structure of F corresponds to that of Fo so that and A(c, e; X) by —RN#F12 Then it follows that F11 S F21 K where F22 - S = F12 — (3.1) LEMMA 3.1. Let 0 k < s, k e N and let Ø(c) be a smooth function. Then we have iÅk4(c) = O (A¯N)• PROOF. Since e; A) = #F12 it is easy to check that A) = 00 -1 + (3.2) e) if Due to Lemma 3.1 our problem is reduced to the problem of constructing an asymptotic null solution for r >< r matrix valued K@, A). PROOF. It is clear that det(F#A) = detF + and hence detF + O(FS) = det Fll det K + 00-1 + ICI)N+I from (3.2). Thus we get the assertion taking N large. From Lemma 3.2, due to the fact that k < s, we obtain that A)] + O(I€IN+I(3.3) Let us (3.4) (3.5) (3.6) (3.7) with some <) where rj are polynomials in define [1 + 00 -1 )] det F I G; A) e) + 00—1)] + and hence Assume that satisfies that is = 0. We consider A-SD; where the equality follows from Lemma 3.3 below. Taking into account the fact that Ko(c, 0) = 0, let us denote the right-hand side of (3.7) by then we have Setting 0) # 0 and In since e) # 0. We now have det Al) (c, e; A) = A-(k+l-r) sc) + 00-1)]. Note that if r = 1 then F ( l ) is scalar and if r 2 2 then k+l—r < s—l because k < s. Now our construction of an asymptotic solution for F@, įS D; A) is reduced to the same problem for a scalar operator or for an r x r (r 2 2) matrix valued operator X) where the assumption of Proposition 3.1 is still verified with s, k and g replaced by s — 1, k + 1 — r and f). We now turn to the other case, that is Fo(c, 0) = 0. We need to improve slightly Lemma 4.3 and Lemma 4.4 in [4]. LEMMA 3.3. Let D) be a differential operator with smooth coeffcients, defined in some open set U and let 0 < k < s, s, k G N. Let 4(c) e COO (U). Then we have e with R@, C; A) = Ej —o C) where Rj@, () are polynomials in with coemcients in COO (U), the sum is finite and G@, + D)) denotes the differential operator with symbol II For later use, we consider also the case when k > s. LEMMA 3.4. Let ô > 0 be a positive number and let GCC, K) be a polynomial in (Œ, \) of degree m. Then we have À ¯S D; À)e G(ÀôŒ, Àô om(œ) + À -S D; À) + À -SD; À). We return to the construction of an asymptotic solution for F. Let us write det F (Œ, 4; À) = [g(Œ, Ç) g; À)] and assume that QI,' satisfies (3.6). Recall (3.7). Taking into account the fact that FO(Œ, = O(À¯I ), let us denote the right-hand side of (3.7) by that is (3.8) From (3.8) it follows that À -m det F O ) (Œ, À) = det F(Œ, + 4); À) + (3.9) because F (Œ, + À) = O(À¯I ). On the other hand, repeating the same arguments as before, one can write det F (Œ, + K); À) = (Œ, Ç) + (3.10) where 0) = 0, 0) # 0. Then we have from (3.9) and (3.10) det g; À) = [dl) (Œ, â) + 00—1)] + À-(k+l-m) Ç) + o(À-l)]• (3.11) In particular this shows that k + 1 m. Summing up, the construction of an asymptotic solution for F (Œ, À ¯S D; À) is reduced to the same problem for the operator verifying the assumption of Proposition 3.1 With k, s and g replaced by k + 1 — m, s — 1 and Ç). This case can not happen more than k times until we arrive at the case det (Œ, À) = g* (Œ, C) + (3.12) with g*@, 0) 0, 0) 0 since the power of A, k, in detF decreases by 1 —m —1 at each step. If (3.12) holds we conclude that 0) # 0, that is the first case, otherwise we would have det = O which is a contradiction, for det Fd@, e) g*@, e). Thus the construction of an asymptotic solution for F@, X) is finally reduced to the same problem for a scalar operator. The construction of an asymptotic null solution for a scalar operator is standard (see [7] or, in the actual framework, [3]). 4. Proof of the Theorem Let A e M(m; 1<0)). we denote A = (All], . . . ,A[m]) where Ali] stands for the i-th column of A. We say that A G 'H(kl, k ) if e H(ki), i = l, . . . , m and we write ö(A) = . . . , )). We say that A is homogeneous if every Atil is a homogeneous polynomial in @, e). LEMMA 4.1. Assume that F e H(kl, Let be the rank of Fo, 0 m— 1, and assume that Fo is homogeneous. Then there ecist M, N and for 1 j m — where M, N and S(j ) are permutation matrices and w?) (c, e), wu+1@, = w@, C) are homogeneous polynomials in G, e) such that G, G = E A-'Gj, 11 where the last m — columns of Go are zero and ö(G) = • • • A( m- u)). PROOF. If = 0 so that Fo = 0 then the assertion is clear. Thus we may assume that 2 1. Let us denote by A(F) 91 ' • • • ' 9%) the p-th submatrix and p-th minor respectively consisting of il ,. . . , i p-th rows and , jp-th columns of F. Since the rank of Fo is p, considering MFN with suitable permutation matrices M, N we may assume that A(FO) 1 Write Fll F12 F21 F22 where Fll = Define with the (u + 1) x (p + 1) matrix W: j)-th cofactor of the (P + 1) x (u + 1) submatrix Fo,ll, 0,11 = so that the first columns of Fo,11W coincide with the first columns of Fo,ll and the last column of Fo,11W is zero. Consider Then we have F* G k ) where k*ß+l When 'J = m — 1 then G = F* and A( l ) = H give the desired matrices because the last column of Go = Fo,11W is zero. Let < m — 1. Conjugating F* on the right with a suitable permutation matrix S we can exchange the ('J + l)-th column and the ('J + 2)-th column of F*. We denote the thus obtained matrix by F( l ) = F#HS. It is clear that F( l ) e H(kl,. . . k kg-{-2' kg-1-3' m Since, by assumption, the rank of Fo is V, we conclude because F(l) = Foils and detH = O. Recalling that one gets We can now apply the same arguments to F ( l). We repeat this procedure m — times. Recalling that F(j - l) e H(kl, k k e-l-j—l' we take vv(j) 11 where WY) is the (g + 1, i)-th cofactor of Fo(jl—1). Consider j the (u + j + l)-th column to the ('J + l)-th column position and the (u + a)-th column to the (u + a + l)-th column position for 1 a j and set F(j) where km), and hence ka + j deg(wß+l). In particular we have det F(j ) e H ka + j deg(wp+l) Thus finally we get where G e H(kl Note that with Go,llFONA( I ) • • • = M Go,21 Go,22 where Go,ll and Go,22 are (u +1) x (g + 1) and (m —P— 1) x (m — 1) submatrix respectively, the last column of Go,ll is zero. Since it follows that A(GO) On the other hand from det Go = w} det Fo the rank of Go is /.J and then we conclude that the last m — columns of Go are zero otherwise we would have rank Go > V. This completes the proof. Let us take the diagonal matrix and consider Then it is clear that F (l ) e H(kfl), , 1€%})) with and hence have DetF = Det F( l ) F) = ö(det A l )). On the other hand from (4.1) it follows that det F ( l ) G H Ekj + (m — u) deg(w) + m LEMMA 4.2. Assume DetF = 00-9, t e N, Fe H(kl' • • • k ) rank Fo = m — 1 such that .,m- l A(FO) 1 II Then there eŒist (j) o 1 (j) 1 w for 1 j t With such that = À -t / m F* —E À-jF* H(kl, . . . , m—l, km + t deg(w) + t), det F* G H E ki + t deg(w) + t wt ô(det F) = À -t ô(det F*). P ROOF. By Lemma 4.1 and the remark after its proof we obtain that there exists A( l ) such that = À-1/mF(1) F(I) = where F ( l ) Q H(kl, km—l' km I). If t — 1 then the proof is complete. If t > 1 then from DetF = À -1 DetF( 1 ) = À -l [det F( I ) + 0(À -s )] = o(À -t ) it follows that det FS I ) = O(À ¯I ) and hence det FS I ) = 0. Since rank FS I ) > m — 1 we conclude that rankF0( 1 ) = m — 1. Noticing l, . . . , m 1 l, . . . , m — l one can repeat the same arguments t times to get the desired assertion. Let us recall where so that F G Recall a!ß! la-i-ßl=rn and (4.2) COROLLARY 4.1. One can write (4.3) for some t e N, 0 < t m. PROOF. Suppose DetF = From Lemma 4.2 we have wm+ l ö(det F) = F* ) with some F* = This shows that ö(det = [hp + o(x-l )] which is a contradiction because ö(det F* ) = 0(1). Since we have proved Theorem 1.1 in [4] when rank Ll (0, en) = m — 2 we may assume that rank Ll (0, en) m — 3. LEMMA 4.3. Let t be as in Corollary 4.1. Then there ecist r, A(j ), 1 S j r such that where DetF= A-t DetF*, det F* e H Evj deg(wj) + E vj , with where the wj(c, G) are homogeneous polynomials in G) and N (j ) S(j ) are permutation matrices. PROOF. Assume that the rank of Fo is 0 m — 3. Then from Lemma 4.1 and the remark following it, there exists A( l ) such that det F ( l ) e H((m — go) deg(wl) + m — #0). We remark that since Fo is a constant matrix then so is A( l ). Then we see that A l ) is homogeneous for 0 j s. Let us set VI = m — 2 3. If VI = t then we have the desired assertion with r = 1. Assume that VI < t. Since WI I DetF = DetF( 1 ) [det F ( 1 ) + = it is clear that det Fo( l ) = 0. Thus one can apply again Lemma 4.1 to F ( 1 ). Note that = rank F( l ) > ,uo. We repeat the same arguments and get F ( 2 ). It is clear that FTP ) is homogeneous for 0 j s— 2. By induction it is easy to see that is homogeneous for 0 j s — p because • • • + O(F S ), and A(.p) are homogeneous. The point of the proof is the following: there exists r s such that rank F (r) = m — 1. Otherwise we would have, setting = rank F(j ) that is vj+l = m — 2 2 for 1 j s and VI 2 3. This is impossible, because from = A — vj/rnF* we have DetF*, where Evj 3 + 2s > 2s+2 2 m. This contradicts (4.3). Assume = m — 1 with some r s so that #A (r) = X— Vj/m p(r) where the conclusion of Lemma 4.1 holds for F* = F Moreover Fo(r) is homogeneous since r s. Noticing DetF(r) = +00-1)], one can apply Lemma 4.2 to conclude that there are Ä(j ), 1 j t with t v • such that where F* verifies This gives and F* verifies the desired assertion with 14+ a = 1, wr+a = w, 1 < a < t. We now prove that Let us consider e—inqmn which yields en + A-q D) + A -q m(c)}. We perform the following dilation: --4 where 0 < ö < 1 will be determined later. Recalling that (1 — ö) — q = —1 — ö — s because q = s + 2, we have Let us recall here an apriori estimate; there exist positive constants C, M, Ä and P e N such that vu e c000 (W), (4.5) To prove (4.4), supposing (4.3) with t < m, we construct an asymptotic solution for F (Ab c, A S D; A) contradicting (4.5). Let A(j ) be as in Lemma 4.3. Setting from Lemma 4.3 it follows that MF(XÖ C, A -Ö - S D; /\ -Ö - S D; A). Recall that F* e H(M, . . . , k*m) where EkJ*vj We construct an asymptotic null solution u: u = exp i with is non trivial in such a way that A(Ä ö c, itself is actually an asymptotic null solution for F (Abc, A). Note that det F* = + • + A m—t) grn—t * -+- • • • deg(Wj). LEMMA 4.4. We have s + 2 t and for 0 i < m — t where hi(c, e) are homogeneous polynomials of degree t + i — q. PROOF. From Lemma 4.3 we have det F* e H(t+d) and hence one can write (4.6) d+t+i—jq20 where C) are homogeneous polynomials of degree d + t + i — jq 2 0. Since ö(det F* ) = by Lemma 4.3 again this shows that the terms g*i0 do not occur in the sum (4.6) for i < m — t. Writing DetF = + A -I gl + . . . + A-e-1) and noting that Det F* = det F* by Lemma 2.4, it follows from Lemma 4.3 that (4.7) (4.8) t+i—jq20 where gij(c, f) are homogeneous polynomials of degree t + i — jq 2 0 and gi0 = 0 for i < m — t as observed above. It follows from go # 0 that t — q 2 0, that is s 2 < t. This is the first assertion. Since t + i — 2q < 0 for 0 i < m — t because m 2(s + 2), no term with j 2 2 occurs in (4.8) and hence the assertion follows from (4.7) with hi = gil• REMARK. Lemma 4.4 implies, in particular, that if we take s m — 2 then we have always Det F = Thanks to Lemma 4.4 and Lemma 4.3 one can write det F* = w[ho + + . . . + where hj (c, C) are homogeneous polynomials of degree t j — q. Let us define ö by (4.9) so that ö 1/2 because s +2 t and q = s + 2. It is clear that —i + (t + i — q)ö (m — t) + mö for 0 i m — t. LEMMA 4.5. There is (j,ä') e nn + l x (Rn \ such that ho(j, (o, €1 ) = 0 has a simple non real root < with 1m < < 0 verifying w(j, C, t') # 0. PROOF. Let us set g = hp + ho and write ho E % (c, fj = 0, j > k, and consider the equation for small e: (4.10) Let us take and replace by in (4.10); this yields cm + fk(c, + = 0. (4.11) Choose (j, K') so that fk(j, t/) # 0 then we can find a simple non real root ( (c, e, e) of (4.11) defined near (j, e = 0 such that [—fk(ö, where we can assume 1m C (x, ( , e) < 0 there because m —k 2 m — Recalling that w is of homogeneous of degree d one can write d and we may assume that Wi = 0, i > io, u.'io 0. Then it is clear that w ex, ex (c, e, e), eel) d—io+aio ('X (C, e', e) t0 Thus choosing O, E') near (j, k') so that and e small (G, cC/) is a desired point and (c, e, e) gives a desired root. Assume that satisfies (4.12) where ( is given in Lemma 4.5. Let us consider e—iÄS+2ö (4.13) Write and apply Lemma 3.4. Then we get e Taking these estimates into account we put and set Then it is clear that Let us define e N by 1 LEMMA 4.6. we have det e; A) = A i6 (c, e) + PROOF. Recall that with G* = we have — A-k+kö k Thus the main contribution of the remainder terms) in the Aexpansion of det G* A) is of the form O (X— pj+ö Pj BovD with some Pi e N which is because ö < 1. On the other hand we see that det We recall that gi (A C, /\ö Øc -k C) = O(Ätö ) for g* e 'H(t + E v,' deg(wj)) and point out that the following inequalities hold for j > m — t for 0 < i < m — t since ö 1/2; this allows us to conclude that (4.14) because is a simple root for g(j, (o, e') = 0 where g = hp + ho. On the other hand it is clear that since m 2s + 2. Then we get the desired assertion setting We are now ready to prove (4.4). Assume that d,' verifies and consider SK R* (c, A) Then it is clear that A) = A - [g f) + where g**@, 0) 0 and 0) # 0. Therefore in order to construct an asymptotic null solution for G**@, A) it suffces to apply Proposition 3.1. Note that, until now we did not choose the initial data for 4, U'. Assume that C') is defined in some open set U x r. From Proposition 3.1 one can find an open set V C U and with which Proposition 3.1 holds. Fix = (jo, j') e V and e r. Then satisfies (4.12) and we can assign initial data for according to: — c') + — d1 2 . This shows that 1m c') 2 — + — c'1 2 ) (4.16) with some c > 0 near with jo. As for e, it satisfies (4.15) and we assign initial data according to: so that 1m "(co, c') C(lco' c'1 2 ) (4.17) with some C > 0 near j. Let be an asymptotic null solution of constructed as above, where Thanks to (4.16) and (4.17) we have 1m A) 2 c' (jo .cq2)'(4.18) BovE with some d > 0 for large near j. In order to exhibit an asymptotic null solution for X) we must still show that A(Aö c, is non trivial. Let us argue by contradiction: suppose that iÄS+2Öqc,Ä) Taking the very special structure of A into account this would imply that which contradicts Lemma 4.5. In view of (4.18), u =—5 — SD; is a desired asymptotic solution for which (4.5) does not hold when —+ oo where e C000 (Rn+ l ) is a cut off function around j. Assuming that DetF = F m [go + go # 0 we next show that a(Det F) = hp. We argue in a way analogous, but simpler, to the above. Let us take ö 0 and construct an asymptotic solution for A-SD; A). The same arguments as in the proof of Lemma 4.3 and Lemma 4.4 show that detF* = g; +A Igl* where f is a homogeneous polynomial of degree m — q. Here we note that if we take s m — 1 then we have always o(Det F) = hp because m — q < 0. Assuming a(Det F) # hp, and necessarily s m — 2, we would have f # 0. We repeat the same arguments as in the proof of Lemma 4.4 to conclude that there is a non real simple root < of hp(j, C, t') 4- f (j, C, t') = 0 with 1m ( < 0 verifying w(j, C, e) # 0. The rest of the proof is a repetition of the preceding arguments. 5. Remarks and examples In this section we prove PROPOSITION 5.1. Let rank LI(O, en) = p and F = [fo + A-I fl + + A-e-l)fs-l]. Denote by GL(m; K (A)) the group of all non singular m x m matrices with entries in K(A) and by SL(m; K(A)) the subgroup of GL(m; K(A)) generated by the matrices Bij (f), i # j, obtained from the unit matrix by replacing the (i, j)-th entry by f. Let us define For A G M (m; K (A)) we set d(A) = A = (Atj). (5.1) (5.2) Multiply F with A on the right; we get BOVE Note that S = and then LEMMA 5.1. Taking N large we have DetK = [1 + 00 -1 )] • DetF. Since K = we write K = F I K*, X) = Considering MK* N, where M, N are suitable permutation matrices, we may assume that We write K* = F ( l ) where 0, ij < 0 and ti 0. It is clear that there exist BS I ) e SL(m; K ( A)) such that 0 where it is easy to see that with 91 = glj and G(.l ) are independent of 1.40 (c) if Zl + j < s + 1. Repeating the same arguments for G( l ), we finally can show that there exist Bi G SL(m; K (A)) such that = diag(x -tl gl, . . . , grn—u where Cj 0 and with DetK = jl • Y2 • • • gm—v• Let us write s—l) * gs—l then it is clear that g; is independent of Lo(c) if j + (Cl + • • • From Lemma 5.1 it follows that DetF = [gö + 98-1] • [1 + 00-1)] where = (m — g) + + (Cl + 12) + + (Cl + . . . + With DetF = [fo + + + If + < — + s + I then we have j + + + ) < s + 1 because ). This proves the assertion. We now give some examples for m = 4 and s — 1. Consider o o o 1o o o o o o o 0o o o o ob o o o where a e R, a > 0 and b e C. Let k = 1 then we see that up to the sign. Since hp = (G — co — ax; Theorem 1.1 shows that b = 0 is necessary for the well posedness of the Cauchy problem. We now take k = 2. Then it is also easy to see that Actually in this case, if a # 1, a > 0 then Ll is strongly hyperbolic and hence any lower order term is allowed (see [10]). BovE References [ 1 ] E. Artin, Geometric Algebra, Interscience Publisher Inc., New York, 1957. [ 2 ] S. Benvenuti, E.Bernardi and A.Bove, The Cauchy problem for hyperbolic systems with multiple characteristics, Bull. Sci. Math., 122 (1998), 603—634. [ 3 ] A. Bove and T. Nishitani, Necessary conditions for the well-posedness of the Cauchy problem for hyperbolic systems, Osaka J. Math. , 39 (2002), 149—179. [ 4] A. Bove and T. Nishitani, Necessary conditions for hyperbolic systems, Bull. Sci. Math. , 126 (2002), 445-479. [ 5] A. D'Agnolo and G. Taglialatela, Sato-Kashiwara determinant and Levi conditions for systems, J. Math. Sci. Univ. Tokyo, 7 (2000), 401—422. [ 6 ] J. Dieudonné, Les déterminants sur un corps non commutatif, Bull. Soc. Math. France, 71 (1943), 27-45. V. Ivrii and V.M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations, Uspekhi Mat. Nauk, 29 (1974), 3—70. [ 8] W. Matsumoto, The Cauchy problem for systems-through the normal form for systems and theory of weighted determinant, Séminaire Equations aux derivées Partielles, 1998—1999, Exp. no. XVIll, 30 pp. Ecole Polytechnique, Palaiseau. 1 9 ] T. Nishitani, Hyperbolicity of localizations, Ark. Mat. , 31 (1993), 377—393. T. Nishitani, Strongly hyperbolic systems of maximal rank, Publ. Res Inst. Math. Sci.. 33 (1997), 765-773. T. Nishitani, Necessary conditions for strong hyperbolicity of first order systems, J. Analyse Math., 61 (1993), 181-229. M. Sato and M. Kashiwara, The Determinant of matrices of pseudo-differential operators, Proc. Japan Acad., 51 (1975), 17—19.