Permanents as formulas of summation over an algebra with a unique n-ary operation Текст научной статьи по специальности «Математика»

УДК 517.55 + 519.1

Permanents as Formulas of Summation over an Algebra with a Unique n-ary Operation

Georgy P. Egorychev*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny 79, Krasnoyarsk, 660041

Russia

Received 22.06.2018, received in revised form 08.09.2018, accepted 04.11.2018 We give a new general definition for permanents over an algebra with a unique n-ary operation and study their properties. In particular, it is shown that properties of these permanents coincide with the basic properties of the classical Binet-Cauchy permanent (1812).

Keywords: permanents, noncommutative and multioperator algebras, the polarization theorem, polynomial identities.

DOI: 10.17516/1997-1397-2018-11-6-796-799.

Introduction

The following concept of the permanent was first introduced independently by J. Binet and A. Cauchy in 1812. The permanent of a square n x n matrix A = (aj) over fields R or C (a commutative ring K) is defined to be the following sum: Per (A) = ^ a'1a{1) x ... ana(n), where

the sum is taken over all permutations of the set {1,2,... ,n}. In other words,

Per(A) = -1 ^ aT(i)ff(i) x ... x aT(n)a(n) = ^2 Sym{aia(i) x ... x ana(n)], (1)

П!

r,aesn o-esn

where Sym{aia(i) x ... x a,na(n)} = -; E aT(1)a-(1) x ... x aT(n)a(n).

n! tesn

Let ^ be a additive commutative semi-group, G an abelian group with division by integers, ^ an algebra with a unique n-ary operation f (x) = f (xi,x2,... ,xn) : ^ G. By analogy with (1) we introduce a more general definition for permanents (compare with [1,2] and many others).

Definition. The e-permanent ePer(A,f) of a square n x n matrix A = (aj) over the algebra ^ is defined to be the following sum

ePer(A,f ) = n f (aT(iMi)'... ,aT(n)a(n)). (2)

T,aES„

In other words,

ePer(A, f) := ^ Sym{f (aMi), .. ., ana(n))} = ^2 Sym{f (aT(i)i, .. ., aT(n)n)}, (3)

aESn tESn

where thе symmetrization operator

Symf (xi,x2, ...,xn):= — ^2 f (x^(i), xa(2), ..., xa(n)) ,

' aeSn

and Symf (xi, x2,..., xn)=f (xi,x2,... ,xn), if f (xi,x2,..., xn) is a symmetric function.

* [email protected] © Siberian Federal University. All rights reserved

1. Results

Properties of permanents ePer(A, f) follow directly from properties of a general term of sums (1)-(2). For example, a general term of sum (3) is a symmetric function of its arguments.

In particular, we have obtained several combinatorial formulas (polynomial identities) for e-permanents that contain several free elements from algebra G. As a special case, one of them includes the known Ryser-Wilf formulas for Per(A) (H. Wilf, 1968; H.Ryser, 1963), which gives the fastest algorithm for computing Per (A). All computation formulas for e-permanents obtained here by means of the known polarization theorem for recovering a polyadditive function from its values on a diagonal [3,4].

Let's denote a algebra such algebra at which f (x1,x2,... ,xn) = 0 for each element from containing though one zero coordinate.

Theorem 1. (a) ePerf (A, f) = ePerf (AT,f).

(b) ePerf (A) is a symmetric function of rows and columns of the matrix A.

(c) If the n-ary operation f (x1,x2,... ,xn): ^ G is polyadditive then ePerf (A) is a polyadditive function of rows and columns of the matrix A.

(d) The e-permanent over the algebra is equal to 0, if the square n x n matrix A contains (up to shifts of rows and columns) a proper a r x k zero-submatrix, where r + k > n. In this case ePer(A, f) = 0, if the matrix A contains a zero-row or a zero-column.

(e) If the algebra contains the unit e, and In is the identity matrix then the equality ePerf (A, f) = e for the e-permanent over is valid.

In general case the Laplace formulas for permanent are not valid.

Theorem 2. If an n-ary operation f (x1,x2,..., xn): ^ G is polyadditive then the following formula for ePer(A, f) over the algebra ^ is valid:

n

ePer(A, f) ^Y}-1^ Y^ Symf (Yl—alj1 -alj2. -a1jk , . . . , Yn-an.1 -anj2. . — anjk ),

i=0 l<ji<...<jk<n

where 71 ,Y2,...,Yn are elements from In a special case, if an n-ary operation

f(xl , x2, ... , xn ): ^ G is symmetric and polyadditive then

n

ePer(A,f) \=YJ(-1)k Y^ f (Yl-a131-a1j2-...-a1jk T ... Tin-ann-anj2-...-anjk ). (4) i=0 1<ji<...<jk<n

Introduce the operation

F (x) := f (x, x,...,x) : ^ ^ G.

Theorem 3. The following formula for ePerf (A, f) over the algebra ^ containing n! any elements Ya G ^0, a G Sn, is valid:

{n k }

T,T,(-1)n-k E F (Ya +E j a(,.. ))/n!. (5)

aeSn k=0 1^ji<...,jk <n s = 1 J

If in (5) Ya = Y for each a G S then

{n k }

J2(-1)n-k(n - k)! F (y + £ ajsis )/n!,

k=0 1^j1<...<jk^n, 1^i1...<ik^n s = 1 I

which for Y = 0 gives

{n k }

J2(-1)n-k(n - k)! Fj2ajs is/n!.

k=1 1^j1<...<jk^n, 1^i1...<ihXn s = 1 J

Remark. In the simplest case, if in (4) we put f (xi,x2,... ,xn)= \\ x^ Rn ^ R, F(x) = xn

i=i

then we get the first polynomial identity for permanents over the field R (G. Egorychev, 1979). In its turn, this polynomial identity for special values of free parametres gives the known Ryser-Wilf formulas (computing algorithm) for the Binet-Cauchy permanents.

It is common knowledge that the determinant det(A) can be computed in poly (n) time. On the other hand, the fastest Ryser-Wilf algorithm known for computing Per (A) runs in n2n~i time. Moreover, L. Valiant (1979) has shown that the problem of computation Per(A) even for (0,1)-matrices is P-complete. The formulas (4)-(5) for ePer(A, f) is the same as the Ryser-Wilf formulas for Per(A) (up to replacing an n-ary operation of multiplicaion for elements of a commutative ring by an n-ary operation f (xi,x2,... ,xn) : ^ G). Thus, the complexity of computation of ePer(A, f) by formulas (4)-(5) directly depends on the complexity of the Ryser-Wilf algorithm for Per(A) and the complexity of (fastest) algorithms for computatin of the concrete polyadditive n-ary operation f (xi,x2,... ,xn) : ^n ^ G and one-ary operation F (x) := f (x, x,... ,x) : ^ ^ G (see, for example, [5]).

Illustrative examples Let Kij, i,i = 1, 2,... ,n, be non-empty convex compact sets in Euclidean space Rn bodies in Rn with the addition of bodies by Minkowski, A be any unimodular transformation in Rn. Let V(Ki, K2,..., Kn) be the mixed volume of bodies Ki,..., Kn, V(K) the volume of body K ( [6], Chapter 4). Then the following formula of multiple summation is valid:

J2 V(AKia(i),...,AKna(n)) =

aeSn

n k \

J2(-1)n-k(n - k)\ J2 V(Ko + £ AKss)/n\,

k=0 i^j1<---<jk ^n, i^i1...<ik ^n s = i I

where K0 is any body in Rn.

Similarly, let Aij, i,j = 1, 2,... ,n be square nxn matrices over the field R, D(Ai, A2,..., An) be the mixed discriminants of matrices Ai,..., An, A be any unimodular n x n matrix over field R ( [6], Sec. 25). Then

J2 D(AKia(i), . . . , AKna(n))} =

aeSn

{n k }

J2(-1)n-k(n - k)\ J2 det(Ao + £ AKjsis j )/n\,

k=0 i^j1<...<jk^n, i^i1...<ik^n s = i I

where A0 is any n x n matrix over the field R.

It is of interest to obtain similar results for Schur fuctions, the mixed discriminants, the resultants, and many other planar and space matrix functions over different algebraic systems of various type and its applications (see [4,7-10] and many others). In our view, an answer to the following question is particularly interesting: for which operations f(xi,x2,... ,xn) : ^ G a fast evalution of ePer(A, f) with the help of quantum computers is possible? (see, example [11]).

I am very grateful to my good collegues L. V. Knaub, S. G. Kolesnikov, V. P. Krivokolesko, A. V. Shchuplev, A. I. Sozutov, V. A. Stepanenko, A. K.Tsikh for their help and useful remarks.

References

[1] A.Barvinok, New permanent estimators via noncommutative determinants, preprint arXiv:math/0007153, 2000, 1-13.

[2] I.M.Gelfand, V.S.Retakh, Determinants of matrices over noncommutative rings, Functional. Anal. and Prilozhen, 25(1991), no. 2, 91-102.

[3] G.P.Egorychev, New formulas for the permanent, Soviet Math. Dokl., 254(1980), no. 4, 784-787.

[4] G.P.Egorychev, A new family of polynomial identities for computing determinants, Math. Dokl., 2(2013), no. 1, 1-3.

[5] V.V.Kochergin, About complexity of computation one-terms of powers, Discrete Analysis, IM SO RAN, Novosibirsk, 27(1994), 94-107 (in Russian).

[6] Yu.D.Burago, V.A.Zalgaller, Geometric Inequalities, N.Y., Springer Verlag, 1988.

[7] G.P.Egorychev, Discrete mathematics. Permanents. Krasnoyarsk, Siberian Federal Univ., 2007 (in Russian).

[8] H.Cartan, Elementary theory of analytic functions of one or several complex variables, N.Y., Dover Publ., 1995.

[9] V.T.Filippov, On the n-Lie algebra of Jacobians, Sibirsk. Math. Zh, 39(1998), no. 3, 660-669 (in Russian).

[10] L.V.Sabinin, Methods of Nonassociative Algebra in Differential Geometry (in Supplement to Russian translation of S.K.Kobayashi and K.Nomizu "Foundations of Differential Geometry"), Moscow, Nauka, 1981 (in Russian).

[11] L.Chakhmakhchyan, N.J.Cerf, R.Garcia-Paton, Quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices // Preprint arXiv: quant-ph. / 1609.02416.2017.1-9.

Перманенты как формулы суммирования над алгеброй с единственной n-арной операцией

Георгий П. Егорычев

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный 79, Красноярск, 660041

Россия

Дано новое общее определение перманентов над алгеброй с единственной n-арной операцией и изучены его свойства. В частности, показано, что свойства этих перманентов совпадают с основными свойствами классического Бине-Коши перманента (1812).

Ключевые слова: перманенты, некоммутативные и мультиоператорные алгебры, теорема поляризации, полиномиальные тождества.