Bounding the rational sums of squares over totally real fields
aa r X i v : . [ m a t h . N T ] J u l BOUNDING THE RATIONAL SUMS OF SQUARES OVERTOTALLY REAL FIELDS
RONAN QUAREZ
Abstract.
Let K be a totally real Galois number field. C. J. Hillar provedthat if f ∈ Q [ x , . . . , x n ] is a sum of m squares in K [ x , . . . , x n ], then f is asum of N ( m ) squares in Q [ x , . . . , x n ], where N ( m ) ≤ [ K : Q ]+1 · ` [ K : Q ]+12 ´ · m ,the proof being constructive.We show in fact that N ( m ) ≤ (4[ K : Q ] − · m , the proof being constructiveas well. Introduction
In the theory of semidefinite linear programming, there is a question by Sturmfels
Question 1.1 (Sturmfels) . If f ∈ Q [ x , . . . , x n ] is a sum of squares in R [ x , . . . , x n ],then is f also a sum of squares in Q [ x , . . . , x n ] ?Hillar ([3]) answers the question in the case where the sum of squares has coef-ficients in a totally real Galois number field : Theorem 1.2 (Hillar) . Let f ∈ Q [ x , . . . , x n ] be a sum of m squares in K [ x , . . . , x n ] where K is a totally real Galois extension of Q . Then, f is a sum of K : Q ] + 1 · (cid:18) [ K : Q ] + 12 (cid:19) · m squares in Q [ x , . . . , x n ] . The aim of this note is to show, modifying a little bit Hillar’s proof, that only(4[ K : Q ] − · m squares are needed (that is Theorem 3.1). Moreover, as in [3],the argument is constructive. 2. Hillar’s method
Having in mind the Lagrange’s four squares Theorem, we focus ourselves on rational sum of squares i.e. linear combination of squares with positive rationnalcoefficients.Let K be a totally real Galois extension of Q which we write K = Q ( θ ) with θ a real algebraic number, all of whose conjugates are also real. We set r = [ K : Q ]and G = Gal( K/ Q ).Let f ∈ Q [ x , . . . , x n ] be a sum of m squares in K [ x , . . . , x n ], namely f = P mk =1 f k , with f k ∈ K [ x , . . . , x n ]. Summing over all actions of G (i.e. “averag-ing”), we get Date : September 26, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Rational sum of squares, semidefinite programming, totally real num-ber field. (1) f = 1 | G | m X k =1 X σ ∈ G ( σf k ) Next, we write each f k in the form f k = r − X i =0 q i θ i where q i ∈ Q [ x , . . . , x n ]. Then,(2) X σ ∈ G ( σf k ) = r X j =1 r − X i =0 q i ( σ j θ ) i ! We may write this sum of squares as the following product of matrices q ... q r − T σ θ . . . ( σ θ ) r − ... ... . . . ...1 σ r θ . . . ( σ r θ ) r − T σ θ . . . ( σ θ ) r − ... ... . . . ...1 σ r θ . . . ( σ r θ ) r − q ... q r − We obtain what is called a Gram matrix (cf [1]) associated to the sum of squaresin (2). Let G = σ θ . . . ( σ θ ) r − ... ... . . . ...1 σ r θ . . . ( σ r θ ) r − T σ θ . . . ( σ θ ) r − ... ... . . . ...1 σ r θ . . . ( σ r θ ) r − Note that the entries of G are in Q since they are invariant under the σ j ’s.Now, we come to the slight modification of the proof of Hillar that will improvethe bound. 3. LU-decomposition of the Gram matrix If u ( x ) denotes the minimal polynomial of the Galois extension K over Q , thenthe ( i, j )-th entry of the matrix G is the i + j − σ , . . . , σ r )the roots of u ( x ). It is well known that the rank of G is equal to r and its signature(the difference between the positive eigenvalues and the negative ones) is equal tothe number of real roots of u ( x ) (see for instance [2, Theorem 4.57]). In our case,we readily deduce that G is a positive definite matrix since K is totally real. Thus,all its principal minors are different from zero (they are strictly positive !) and G admits a LU-decomposition which we may put in a symmetric form G = U T DU where D is diagonal and U is upper triangular with diagonal identity, and U, D have rational entries.We may view this decomposition as a matricial realization of the Gauss algorithmwhich reduce the quadratic form given by G . OUNDING THE RATIONAL SUMS OF SQUARES OVER TOTALLY REAL FIELDS 3
Now, if we denote by f , . . . , f r the polynomials in Q [ x , . . . , x r ] appearing asthe rows of the matrix U × q ... q r − and by d , . . . , d r the rational entries ontothe diagonal of D , then we get from (2) the identity :(3) 1 | G | X σ ∈ G ( σf k ) = d | G | g + . . . + d r | G | g r This construction leads to
Theorem 3.1.
Let f ∈ Q [ x , . . . , x n ] be a sum of m squares in K [ x , . . . , x n ] , where K is a totally real Galois extension of Q . Then, f is a sum of (4[ K : Q ] − · m squares in Q [ x , . . . , x n ] .Proof. By (1) and (3), it suffices to apply Lagrange’s four squares Theorem to getthat f is a sum of 4[ K : Q ] · m squares in Q [ x , . . . , x n ].But let us note that the first diagonal entry of D is always d = r = [ K : Q ].Then, by the averaging process the first coefficient appearing in the rational sum ofsquares in (3) is d | G | = 1 : already a square in Q ! Whereas the others coefficients d i | G | in the rational sum of squares could be any positive rational which we rewriteas a sum of 4 squares. This concludes the proof. (cid:3) Remark . Beware that if we perform the Cholesky algorithm to the matrix G instead of the LU-decomposition, it yields a factorisation G = U T U where U islower triangular but with entries in Q [ √ d , . . . , √ d r ] for some integers d , . . . , d r .Then, an averaging argument would produce identities over Q but will raise thenumber of squares by an unexpected multiplicative factor 2 [ K : Q ] .Let us consider as an example, the simple case of quadratic extensions : Example . Let K = Q ( √ d ) where d ∈ Q is not a square. The extension K isalways Galois, and it is totally real if d ≥ f ∈ Q ( x , . . . , x n ) be such that f = P mk =1 ( a k + b k √ d ) with a k , b k ∈ Q ( x , . . . , x n ). Since f has rational coefficients, by averaging we get f = 12 m X k =1 ( a k + b k √ d ) + ( a k − b k √ d ) = m X k =1 ( a k + db k )It remains to write d as a sum of l ≤ f is asum of at most (1 + l ) · m squares in Q ( x , . . . , x n ).As another illustration, we apply our method to [3, Example 1.7] : Example . Consider the polynomial f = 3 − y − x + 18 y + 3 x + 12 x y − xy + 6 x y which is the following sum of squares f = ( x + α y + βxy − + ( x + β y + γxy − + ( x + γ y + αxy − in Q ( α )[ x, y ] where α, β, γ are the real roots of the polynomial u ( x ) = x − x + 1. RONAN QUAREZ
We do not need to average and directly compute the matrix G and its symmetricLU-decompositon − − = − − Because of the relations β = 2 − α − α and γ = α −
2, the vector of polynomials q = ( q , q , q ) T is q = ( x + 2 xy − , − xy , y − xy ) T and hence f = 3 (cid:16) ( x + 2 xy −
1) + 2 ( y − xy ) (cid:17) +6 (cid:18) − xy −
12 ( y − xy ) (cid:19) + 92 (cid:16) y − xy (cid:17) a rationnal sum of 3 squares, to compare with the rational sum of 6 squaresobtained in [3]. References [1] M.D. Choi, T.Y. Lam, B. Reznick,
Sums of squares of real Polynomials , Proc. Sympos.Pure Math., 58, Part 2, Amer. Math. Soc., Providence, RI, 1995.[2] S. Basu, R. Pollack, M.F. Roy,
Algorithms in Real Algberaic Geometry , Springer[3] C. J. Hillar,
Sums of squares over totally real fields are rational sums of squares , Proc.Amer. Math. Soc. 137 (2009), no. 3, 921–930.
IRMAR (CNRS, URA 305), Universit´e de Rennes 1, Campus de Beaulieu, 35042 RennesCedex, France
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