On a matrix function interpolating between determinant and permanent

On a matrix function interpolating between determinant and permanent Emmanuel Briand Departamento Matemáticas, Facultad de Ciencias, Estadística y Computación, Universidad de Cantabria, Avenidad de Los Castros s/n, 39005 Santander, Spain Abstract A matrix function, depending on a parameter t, and interpolating between the determinant and the permanent, is introduced. It is shown this function admits a simple expansion in terms of determinants and permanents of sub-matrices. This expansion is used to explain some formulas occurring in the resolution of some systems of algebraic equations. AMS classification: 05E05; 13P10; 15A15 Keywords: Matrix functions; Multisymmetric functions; Zero-dimensional algebraic systems 1. A function interpolating between permanent and determinant Given a permutation σof {1,...,r}, denote with O(σ) the set of its orbits, and define the following class function with parameter: γt(σ) = ω∈O(σ) (1−t|ω|). Here |ω|is the cardinality of ω. One can now consider the following function, defined upon the square matrices of order r: t(M) = σ∈Sr γt(σ)  i mi,σ(i).(1) The object of this note is the following identity: Theorem 1 t(M) =(−t)|I|det M[I]per M[I∗],(2) where the sum is carried over the subsets Iof {1,...,r}. The notation M[I]refers to the submatrix of Mobtained by keeping only the lines with index in I, and as well the columns with index in I;andI∗is the complement of I. Proof. Expanding the product γt(σ) = ω∈O(σ) (1−t|ω|), one gets  U⊂O(σ) (−t)ω∈U|ω|. Associate to each subset Uof O(σ ) the union Iof the orbits that are elements of U. This way each subset stabilized by σis obtained exactly once. The permutation σ induces a permutation σIof Iand a permutation σI∗of its complement; the signature of σIis (−1)|I|−|U|. Thus the product is also equal to  I|σ(I)=I (−t)|I|ε(σI). Using this in the formula (1)wheret(M) is defined, it comes:  σ∈Sr,I|σ(I)=I (−t)|I|ε(σI) i mi,σ(i), which decomposes into  σ∈Sr,I |σ(I)=I (−t)|I|ε(σI) i∈I mi,σI(i)  i∈I∗ mi,σI∗(i) = I (−t)|I|det M[I]per M[I∗]. And this is the announced result.  The function tinterpolates between the permanent and the determinant because: •when t=0, that is: when considering the constant term, one obtains the permanent. •when t→−∞, that is: when considering the leading coefficient, that of tn, one obtains the determinant. Another interesting specialization is obtained with t=1; it gives an identity due to Muir [1]: 0= I (−1)|I|det M[I]per M[I∗]. Yet another specialization, t=1/2, appears in the study of certain systems of algebraic equations, where the identity (2) happens to be useful. This is what we expose in the next part. 2. An application to some systems of polynomial equations Let a system of rquadratic equations F1=···=Fr=0inrunknowns X1,..., Xr, with the following particular shape: Fi=X2 i+ r  j=1 ui,j Xj+ci. Such a system has always finitely many roots, namely 2rif each root is counted as many times as indicated by its multiplicity. We will denote with Uthe matrix of the uij . One wants to calculate the multisymmetric power sums of the multiset of the roots of this system 1, denote them with pαfor α∈Nr. This means that pαis the sum of the evaluations of the monomial Xα1 1···Xαr rat the roots (repeated when there are multiplicities). The power sums which may be difficult to obtain are the first ones: the pαwith αwith 0 – 1 coordinates 2). These first power sums may be computed using the identities of Aizenberg and Kytmanov [3]3:  α∈Nr pα Xα=SX1···XrJac F1···Fr .(3) In the right-hand side of the formula, Jac is the Jacobian of F1,...,F r;theSsymbol means that a series expansion, following the decreasing total degree, is applied to 1Such analogues of the symmetric functions where introduced by several authors at the end of the nineteenth century. See [2] for more references and specially connections to systems of polynomial equations with finitely many solutions. 2The other power sums are easily deduced from the first ones. For instance, the first equation of the system yields the inductive formula p(α1+2),α2,...,αr+ j u1,j pα1,...,(1+αi),...,αr+c1pα=0. 3Such identities were first written by Jacobi, in [4]. the rational fraction (this is possible because the denominator has exactly one term of maximal total degree, that is: X2 1···X2 r), and next that all the Laurent monomials with numerator are removed from this expansion (symbol). Applying this to the first examples of the considered systems, one finds that •for r=1, p1=−u11, •for r=2, p11 =u11u22 +3u12u21, •for r=3, p111 =−u11u22u33 −3u11u23u32 −3u13u22u33 −3u12u21u33 −7u12u23u31 −7u13u21u32. These observations, and a few next ones, lead to conjecture that p11···1=(−1)r σ∈Sr ω∈O(σ) (2|ω|−1) i ui,σ(i).(4) Making use of the identity (2), we will prove the above formula, and the more general one: Theorem 2 pA=(−1)|A|2n−|A| σ∈SA ω∈O(σ) (2|ω|−1) i∈A ui,σ(i).(5) Here was made the following abuse of notation: given a subset Aof {1,...,n},it was also denoted with Athe vector whose ith coordinate is 1 if i∈A,0ifi∈ A. Proof. It is convenient to re-write the quotient under the form: Jac/(X1···Xr) (F1/X2 1)···(Fr/X2 r), so that each of its term lies in the sub-algebra of Laurent monomials generated by the 1/Xiand the Xi/X2 j; that is on the one hand Jac X1···Xr = I 2n−|I|det U[I] XI, and in the other Fi X2 i =1+ j ui,j Xj X2 i +ci X2 i . Now we may work modulo the ideal generated by the 1/X2 iand the (Xj/X2 i)2,since we are only interested in the monomials with shape 1/XA, which all lie outside this ideal. One has then Fi X2 i ≡1+ j ui,j Xj X2 i ≡ j1+ui,j Xj X2 i which has as inverse  j1−ui,j Xj X2 i. Thus it remains to compute the terms in 1/XAin the expression:  I 2n−|I|det U[I] XI i,j 1−ui,j Xj X2 i, and in this aim, it is now sufficient to work modulo the module over the polynomials in 1/X1,...,1/Xrgenerated by the Xj/X2 ifor i/=j. One has then:  i,j 1−ui,j Xj X2 i≡ I (−1)|I|per U[I] XI, because the XJ/(XI)2that appear in the expansion of the product are in the submodule unless I=J. Finally the coefficient of 1/XAis given by  I⊂A 2n−|I|det U[I](−1)|A|−|I|per U[A\I], and one recognizes in it (−1)|A|2n1/2(U[A]), which re-writes easily as formula (5).  References [1] T. Muir, A relation between permanents and determinants, Proc. Roy. Soc. Edinburgh 22 (1897) 134–136. [2] E. Briand, Polynômes multisymétriques, Thèse de doctorat, Université de Rennes 1 et Universidad de Cantabria, 2002. [3] L.A. A˘ızenberg, A.M. Kytmanov, Multidimensional analogues of Newton’s formulas for systems of nonlinear algebraic equations and some of their applications, Sibirsk. Mat. Zh. 22 (2) (1981) 19–30; English transl. in Sib. Math. J. 22 (1981) 180–189. [4] C. Jacobi, Theoremata nova algebrica systema duarum aequationum inter duas cariabiles propositarum, J. Reine Angew. Math. 14 (1835) 281–288 [in Latin].