About Necessary and Sufficient Condition for Strong Stationarity of the Positive Quadratic Form

In the paper the finite criteria of the strong stationarity is given for the positive quadratic forms. Before known criteria requires of infinite number of equalities.

The set of all points of the space N R which correspond to positive quadratic forms we denote by N K .
Suppose the positive quadratic form of n variables (p.q.f.) Is given with real coefficients , and is the group of integer valued automorphisms of the form f .P.q.f.f of the form (1) is called [1] perfect form (p.f.) if it is determined by its arithmetic minimum ) ( f m and representations (2), i.e. if the following system has unique solution with respect to coefficients ij a : Let s be complex number; we consider Epstein zeta- It is said [3], that p.q.f.f is final extremal, if there exists the number , 1 0 ≥ s such that for all 0 s s > the form f is s -extremal .
satisfying the following conditions is called [3] sstationary: where The concept of perfect, final extremality and strong stationarity of p.q.f are connected with each other by the following [3].
Delone-Rizhkov Theorem.[3] P.q.f.f is final exstremal if and only if when it is perfect and strong stationary.
Let arbitrary finite group } {g of integer valued unimodular form , which transforms itself with all matrix of the group } {g , be given.

Since the quadratic form
, in transformation with matrix } { ik g g = passes to the quadratic form with coefficients , then if g transforms the form f itself ( g is the integer valued automorphism of the f ), we get the following equality ) .
Here we obtain ) and passes through vertex O of the cone N K .But, according to Mashke's theorem [4], for any finite group of integer valued unimodular matrix there exists positive quadratic form, which transforms to itself by all matrixes of this group.Therefore the manifold which considered above is called [4] Brave manifold, corresponding to the given group } {g , and is denoted by .In the case, when In particular, where is action of the automorphism g to real quadratic form of the form (1), ) ,..., ( It should be noted that equalities (4) and ( 5) are equivalent.Indeed, assuming in , using (4), we get (5).And inversely, multiplying both parts of ( 5) by ii a and ij a 2 , and summing over , we obtain (4).In the work [5] S.S.Rizhkov posted the problem of finding finite necessary and sufficient conditions of strong stationarity, because existing such conditions require checking of infinite number equalities [5].
Here we solve this problem in some sense.

The main result
Theorem.Assume Brave manifold } {g G

В
with respect to integer valued automorphisms of p.q.f.
is point set.Then in order to 0 f becomes strong stationary it is necessary and sufficient fulfilling of the following conditions Proof.Necessity.Suppose p.q.f.0 f is strong stationary.Therefore, it is s stationary and taken place the equality (3) (see [3]).Applying the group ) ( 0 f G to both parts of equality (3) and using (5), consequently we get Necessity of the Theorem is proved.
Sufficiency.Assume the equalities ( 5) and ( 6) are valid.Then, doing all things in inverse order of the proof of necessity of the theorem, starting with (10) and finishing (7), we get (3).Really, from (10) using ( 5) we have Multiplying both parts of (11) by The last equality is equivalent to (3), or the series in the left and the right parts of the equality differ from each other at most permutation of the terms.Therefore Hence, 0 f is s stationary.The conditions (2) and ( 6) do not depend on parameter s .
it is easy to write the coefficients 0 ij с which in the work [6] were calculated by Computer.

В
is the collection of all that points of the cone N K , every of which transforms to itself by that finite subgroup } {g of the equivalence group } {G , i.e. of the group all integer valued unimodular matrixes ( n n ⋅ ) .

1 Introduction. Statement of the problem
linear homogeneous equations with respect to coefficients ij a of the form f .If the form f transforms to itself by any transformation of the , m remain arbitrarily, and other coefficients are expressed in them by Kramer's formula.Therefore in the space of coefficients of quadratic forms it is obtained linear manifold.This monofold is m -dimensional (where N m≤ ≤ 1