## SUMS OF SQUARES OVER TOTALLY REAL FIELDS ARE RATIONAL SUMS OF SQUARES

Citations: | 1 - 0 self |

### BibTeX

@MISC{Hillar_sumsof,

author = {Christopher J. Hillar},

title = {SUMS OF SQUARES OVER TOTALLY REAL FIELDS ARE RATIONAL SUMS OF SQUARES},

year = {}

}

### OpenURL

### Abstract

Abstract. Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1,..., xn] is a sum of m squares in K[x1,..., xn], then f is a sum of 4m · 2 [L:Q]+1([L: Q] + 1 2 squares in Q[x1,..., xn]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems. 1.

### Citations

5102 | Matrix Analysis - Horn, Johnson - 1985 |

241 | Semidefinite programming relaxations for semialgebraic problems
- Parrilo
(Show Context)
Citation Context ...or finding representations of positive semidefinite polynomials as sums of squares. These algorithms have many applications in optimization, control theory, quadratic programming, and matrix analysis =-=[18, 19, 20, 21, 22]-=-. For a noncommutative application of these techniques to a famous trace conjecture, see the papers [1, 8, 13, 16] which continue on the work of [9]. One major drawback with these algorithms is that t... |

132 |
Introduction to quadratic forms over fields
- Lam
- 2005
(Show Context)
Citation Context ...18] that if f ∈ Q[x1, . . .,xn] is a sum of squares of rational functions in R(x1, . . . , xn), then it is a sum of squares in Q(x1, . . . , xn). Moreover, it is known that 2 n+2 such squares suffice =-=[10]-=-. However, the transition from rational functions to polynomials is often a very delicate one. For instance, not every polynomial that is a sum of squares of rational functions is a sum of squares of ... |

72 | Symmetry groups, semidefinite programs, and sums of squares
- Gatermann, Parrilo
(Show Context)
Citation Context ...ediately from Theorem 2.1 and Lagrange’s four square theorem (every positive rational number is the sum of at most four squares). □ Remark 2.3. This averaging argument can also be found in the papers =-=[3, 6]-=-. We will focus our remaining efforts, therefore, on proving Theorem 2.1.SUMS OF SQUARES OVER TOTALLY REAL FIELDS 5 3. Vandermonde Factorizations To prepare for the proof of Theorem 2.1, we describe ... |

45 | Introducing SOSTOOLS: A General Purpose Sum of Squares Programming Solver
- Papachristodoulou, Parrilo, et al.
- 2002
(Show Context)
Citation Context ...or finding representations of positive semidefinite polynomials as sums of squares. These algorithms have many applications in optimization, control theory, quadratic programming, and matrix analysis =-=[18, 19, 20, 21, 22]-=-. For a noncommutative application of these techniques to a famous trace conjecture, see the papers [1, 8, 13, 16] which continue on the work of [9]. One major drawback with these algorithms is that t... |

44 |
Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra
- Prestel, Delzell
- 2001
(Show Context)
Citation Context ... a Gram matrix for f is positive semidefinite with rational entries, then f is a sum of rational squares by a generalization of Lagrange’s theorem for matrices [7]. It follows from a theorem of Artin =-=[18]-=- that if f ∈ Q[x1, . . .,xn] is a sum of squares of rational functions in R(x1, . . . , xn), then it is a sum of squares in Q(x1, . . . , xn). Moreover, it is known that 2 n+2 such squares suffice [10... |

35 |
Minimizing polynomial functions, Algorithmic and quantitative real algebraic geometry
- Parrilo, Sturmfels
(Show Context)
Citation Context ...or finding representations of positive semidefinite polynomials as sums of squares. These algorithms have many applications in optimization, control theory, quadratic programming, and matrix analysis =-=[18, 19, 20, 21, 22]-=-. For a noncommutative application of these techniques to a famous trace conjecture, see the papers [1, 8, 13, 16] which continue on the work of [9]. One major drawback with these algorithms is that t... |

33 | An algorithm for sums of squares of real polynomials
- Powers, Wörmann
- 1998
(Show Context)
Citation Context ...ial is a sum of real polynomial squares if and only if it can be written in the form (1.1) f = v T Bv, in which v is a column vector of monomials and B is a real positive semidefinite (square) matrix =-=[25]-=-; in this case, the matrix B is called a Gram matrix for f. If B happens to have rational entries, then f is a sum of squares in Q[x1, . . . , xn] (this follows from a Cholesky factorization argument ... |

31 |
The Pythagoras number of some affine algebras and local algebras
- Choi, Dai, et al.
- 1982
(Show Context)
Citation Context ...and R ⊗Q K. We do not know how much the factor 2 [L:Q]+1( ) [L:Q]+1 2 can be improved upon, although we suspect that for polynomial rings, it can be improved substantially. We remark that it is known =-=[2]-=- that arbitrarily large numbers of squares are necessary to represent any sum of squares over R[x1, . . . , xn], n > 1, making a fixed bound (for a given n) as in the rational function case impossible... |

30 | Uniform denominators in Hilbert’s Seventeenth Problem - Reznick - 1995 |

18 | The algebraic degree of semidefinite programming
- Nie, Ranestad, et al.
- 2006
(Show Context)
Citation Context ...one. For instance, not every polynomial that is a sum of squares of rational functions is a sum of squares of polynomials [14, p. 398]. More generally, Sturmfels is interested in the algebraic degree =-=[17]-=- of maximizing a linear functional over the space of all sum of squares representations of a given polynomial that is a sum of squares. In the special case of Question 1.1, a positive answer signifies... |

15 |
Advances on the Bessis-Moussa-Villani Trace Conjecture
- Hillar
- 2007
(Show Context)
Citation Context ...c programming, and matrix analysis [18, 19, 20, 21, 22]. For a noncommutative application of these techniques to a famous trace conjecture, see the papers [1, 8, 13, 16] which continue on the work of =-=[9]-=-. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In this regard, Sturmfels has asked w... |

14 |
Quadratic forms under algebraic extensions
- Elman, Lam
- 1976
(Show Context)
Citation Context ...er field with Galois closure L. If f ∈ R is a sum of m squares in R ⊗Q K, then f is a sum of 4m · 2 [L:Q]+1( ) [L:Q]+1 2 squares over R. Remark 1.6. One can view Theorem 1.5 as a “going-down” theorem =-=[4]-=- for certain quadratic forms over the rings R and R ⊗Q K. We do not know how much the factor 2 [L:Q]+1( ) [L:Q]+1 2 can be improved upon, although we suspect that for polynomial rings, it can be impro... |

13 |
Even symmetric sextics
- Choi, Lam, et al.
- 1987
(Show Context)
Citation Context ...ediately from Theorem 2.1 and Lagrange’s four square theorem (every positive rational number is the sum of at most four squares). □ Remark 2.3. This averaging argument can also be found in the papers =-=[3, 6]-=-. We will focus our remaining efforts, therefore, on proving Theorem 2.1.SUMS OF SQUARES OVER TOTALLY REAL FIELDS 5 3. Vandermonde Factorizations To prepare for the proof of Theorem 2.1, we describe ... |

11 |
la représentation en somme de carrés des polynômes á une indéterminée sur un corps de nombres algébriques
- Pourchet, Sur
- 1971
(Show Context)
Citation Context ...this true even if B is not invertible? We suspect not. Although the general case seems difficult, Question 1.1 has a positive answer for univariate polynomials due to results of Landau [15], Pourchet =-=[24]-=-, and (algorithmically) Schweighofer [29]. In fact, Pourchet has shown that at most 5 polynomial squares in Q[x] are needed to represent every positive semidefinite polynomial in Q[x], and this is bes... |

10 |
Proof of the cases p ≤ 7 of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture
- Hägele
- 2007
(Show Context)
Citation Context ...ons in optimization, control theory, quadratic programming, and matrix analysis [18, 19, 20, 21, 22]. For a noncommutative application of these techniques to a famous trace conjecture, see the papers =-=[1, 8, 13, 16]-=- which continue on the work of [9]. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In ... |

10 | Sums of Hermitian squares and the BMV conjecture
- Klep, Schweighofer
(Show Context)
Citation Context ...ons in optimization, control theory, quadratic programming, and matrix analysis [18, 19, 20, 21, 22]. For a noncommutative application of these techniques to a famous trace conjecture, see the papers =-=[1, 8, 13, 16]-=- which continue on the work of [9]. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In ... |

10 |
Exploiting algebraic structure in sum of squares programs
- Parrilo
(Show Context)
Citation Context |

9 |
die Darstellung definiter Funktionen durch Quadrate
- Landau, Uber
- 1906
(Show Context)
Citation Context ...m matrix B. Is this true even if B is not invertible? We suspect not. Although the general case seems difficult, Question 1.1 has a positive answer for univariate polynomials due to results of Landau =-=[15]-=-, Pourchet [24], and (algorithmically) Schweighofer [29]. In fact, Pourchet has shown that at most 5 polynomial squares in Q[x] are needed to represent every positive semidefinite polynomial in Q[x], ... |

9 |
On D. Hägeles approach to the Bessis-Moussa-Villani conjecture, preprint
- Landweber, Speer
(Show Context)
Citation Context ...ons in optimization, control theory, quadratic programming, and matrix analysis [18, 19, 20, 21, 22]. For a noncommutative application of these techniques to a famous trace conjecture, see the papers =-=[1, 8, 13, 16]-=- which continue on the work of [9]. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In ... |

6 |
A Macaulay 2 package for computing sum of squares decompositions of polynomials with rational coefficients
- Peyrl, Parrilo
(Show Context)
Citation Context ...1.1 is not known, Parrilo and Peyrl have written an implementation of SOSTOOLS in the algebra package Macaulay 2 that attempts to find rational representations of polynomials that are sums of squares =-=[23]-=-. Their idea is to approximate a real Gram matrix B with rational numbers and then project back to the linear space of solutions governed by equation (1.1). The following result says that Question 1.1... |

5 | Sums of Hermitian Squares as an Approach to the BMV Conjecture, preprint http://arxiv.org/abs/0802.1153
- Burgdorf
(Show Context)
Citation Context |

5 |
Expressing a polynomial as the characteristic polynomial of a symmetric matrix
- Fiedler
- 1990
(Show Context)
Citation Context ... ⎦ .SUMS OF SQUARES OVER TOTALLY REAL FIELDS 7 If we allow A to contain square roots of rational numbers, however, then there is always such a symmetric A. This is the content of a result of Fiedler =-=[5]-=-. We include his proof for completeness. Theorem 4.2 (Fiedler). Let u(x) ∈ C[x] be monic of degree r and let b1, . . . , br be distinct complex numbers such that u(bk) ̸= 0 for each k. Set v(x) = ∏r k... |

5 | An elementary and constructive solution to Hilbert’s 17th problem for matrices
- Hillar, Nie
(Show Context)
Citation Context ...appens to have rational entries, then f is a sum of squares in Q[x1, . . . , xn] (this follows from a Cholesky factorization argument or from a matrix generalization of Lagrange’s four square theorem =-=[10]-=-). Thus, in the language of quadratic forms, Sturmfels is asking whether the existence of a positive semidefinite Gram matrix for f ∈ Q[x1, . . . , xn] over the reals implies that one exists over the ... |

4 |
Algorithmische Beweise für Nichtnegativ- und Positivstellensätze, Diplomarbeit an der Universität Passau
- Schweighofer
- 1999
(Show Context)
Citation Context ... suspect not. Although the general case seems difficult, Question 1.1 has a positive answer for univariate polynomials due to results of Landau [15], Pourchet [24], and (algorithmically) Schweighofer =-=[29]-=-. In fact, Pourchet has shown that at most 5 polynomial squares in Q[x] are needed to represent every positive semidefinite polynomial in Q[x], and this is best possible. It follows from Artin’s solut... |

3 |
Sums of Hermitian squares and the BMV conjecture, preprint
- Klep, Schweighofer
(Show Context)
Citation Context ...ons in optimization, control theory, quadratic programming, and matrix analysis [12, 13, 14, 15, 16]. For a noncommutative application of these techniques to a famous trace conjecture, see the papers =-=[5, 9]-=- which build on the work of [6]. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In thi... |

2 | A real symmetric tridiagonal matrix with a given characteristic polynomial
- Schmeisser
- 1993
(Show Context)
Citation Context ... Since the monic polynomial det(xI −A) and u(x) agree for x = b1, . . . , br, it follows that they are equal. □ Remark 4.3. There are simpler, tridiagonal matrices which can replace the matrix A (see =-=[28]-=-); however, square roots are still necessary to construct them. The following corollary allows us to form a real symmetric matrix with characteristic polynomial equal to the minimal polynomial for θ o... |

1 |
Some identities for elements of a symmetric matrix
- Ilyusheckin
(Show Context)
Citation Context ...MS OF SQUARES OVER TOTALLY REAL FIELDS 5 3. Vandermonde Factorizations To prepare for the proof of Theorem 2.1, we describe a useful matrix factorization. It is inspired by Ilyusheckin’s recent proof =-=[12]-=- that the discriminant of a symmetric matrix of indeterminates is a sum of squares, although it would not surprise us if the factorization was known much earlier. Let A = AT be an r × r symmetric matr... |

1 |
R.Elman,T.Y.Lam,Quadratic forms under algebraic extensions
- Ann
- 1976
(Show Context)
Citation Context ...number field with Galois closure L. Iff∈ R is a sum of m squares in R ⊗Q K, thenfis a sum of 4m · 2 [L:Q]+1( ) [L:Q]+1 2 squares over R. Remark 1.6. One can view Theorem 1.5 as a “going-down” theorem =-=[4]-=- for certain quadratic forms over the rings R and R ⊗Q K. We do not know how much the factor 2 [L:Q]+1( ) [L:Q]+1 2 can be improved upon, although we suspect that for polynomial rings, it can be impro... |