## Some Concrete Aspects Of Hilbert's 17th Problem (1996)

Venue: | In Contemporary Mathematics |

Citations: | 98 - 4 self |

### BibTeX

@INPROCEEDINGS{Reznick96someconcrete,

author = {Bruce Reznick},

title = {Some Concrete Aspects Of Hilbert's 17th Problem},

booktitle = {In Contemporary Mathematics},

year = {1996},

pages = {251--272},

publisher = {American Mathematical Society}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper is dedicated to the memory of Raphael M. Robinson and Olga Taussky-Todd. 1. Introduction

### Citations

536 |
Lie algebras
- Jacobson
- 1979
(Show Context)
Citation Context ... same degree, then (f=g)(t) can be written as above, but where f jk and g jk are positive definite polynomials of the same degree. There are many expositions of Artin's proof in the literature, e.g., =-=[5, 47, 49, 50]-=-. Ribenboim [74] and Pfister [61] wrote surveys on Hilbert's 17th Problem in the 1970s; two more recent surveys are by Gondard [30] and Scheiderer [78]. The deep connections of Hilbert's 17th Problem ... |

164 |
Géométrie Algébrique Réelle
- Bochnak, Coste, et al.
- 1987
(Show Context)
Citation Context ... same degree, then (f=g)(t) can be written as above, but where f jk and g jk are positive definite polynomials of the same degree. There are many expositions of Artin's proof in the literature, e.g., =-=[5, 47, 49, 50]-=-. Ribenboim [74] and Pfister [61] wrote surveys on Hilbert's 17th Problem in the 1970s; two more recent surveys are by Gondard [30] and Scheiderer [78]. The deep connections of Hilbert's 17th Problem ... |

95 |
The Theory of Spherical and Ellipsoidal Harmonics
- Hobson
- 1955
(Show Context)
Citation Context ... variable, then h(D)F (Gn ) = X k0 2 d 2 2k k! \Delta k (h)F (d\Gammak) (Gn ): The right-hand side is a finite sum; if k ? d=2, then \Delta k (h) = 0. Hobson's Theorem is proved in [43, 44], see also =-=[45]; it was l-=-auded by Hardy [34] as an "elegant theorem in formal differentiation". Now, set F (t) = t s , so F (j) (t) = (s) j t s\Gammaj : h(D)G s n = X k0 (s) d\Gammak 2 2k\Gammad k! \Delta k (h)G s\G... |

65 |
Über die Darstellung definiter Formen als Summe von Formenquadraten”, em
- Hilbert
- 1933
(Show Context)
Citation Context ...2t 2 \Gamma 1) 2 = \Gamma t 3 \Gamma 1 2 t \Sigma p 3 2 \Delta 2 + \Gamma t 2 \Upsilon p 3 2 t \Gamma 1 2 \Delta 2 : In 1888, the 26-year old David Hilbert proved two remarkable results in one paper, =-=[38]. First, he sho-=-wed that \Sigma 3;4 = P 3;4 ; in fact, he showed that every p 2 P 3;4 can be written as the sum of three squares of quadratic forms. (For an elementary proof, with "three" replaced by "... |

56 |
sums of even powers of real linear forms
- Reznick
- 1992
(Show Context)
Citation Context ...in Gel'fand-- Vilenkin [29, pp. 232--235], which also established the connection between forms in \Delta n;m and the Hamburger moment problem in n \Gamma 1 variables. For more on this connection, see =-=[71]-=- and the references contained within. Robinson [77] made a judicious choice of OE and / that greatly simplified Hilbert's methods (see x4b), and cited an unpublished example of Ellison using the origi... |

49 |
Sums of squares of real polynomials
- Choi, Lam, et al.
- 1995
(Show Context)
Citation Context ... inequality, and the fact that these forms are not sos follows in a manner similar to that shown in x4a for M . Choi and Lam called this approach the "terminspection method"; it was later ge=-=neralized [19] as the &q-=-uot;Gram matrix method", which is described in x5b. Choi and Lam constructed a number of other examples of psd forms that are not sos. One is a symmetric quaternary quartic: P x 2 i x 2 j + P x 2... |

39 |
Über die Zerlegung definiter Funktionen in
- Artin
- 1927
(Show Context)
Citation Context ... of four squares. Elements that are not totally positive, such as p 2 above, are negative in at least one embedding into R, and so cannot be sums of squares at all. c. After 1920. In 1927, Emil Artin =-=[1]-=- used the Artin-Schreier theory of real closed fields to answer Hilbert's 17th Problem in the affirmative, under the additional hypothesis that the field K has a unique order. (This hypothesis is sati... |

30 |
Uniform denominators in Hilbert’s Seventeenth Problem
- Reznick
- 1995
(Show Context)
Citation Context ...bove, and a detailed exposition of my recent contribution towards understanding the second question above for positive definite forms. The paper on which I based the second part has appeared in print =-=[72]-=-, and has also been discussed in a recent Monthly article [64]. For these reasons, this paper mainly addresses the first question. An earlier, unrefereed, version of this manuscript appeared [73] in t... |

29 |
Extremal psd forms with few terms
- Reznick
- 1978
(Show Context)
Citation Context ...amma1 \Gamma nu 2 )t 2 1 \Delta \Delta \Delta t 2 n\Gamma1 + u 2n 2 \Delta n;2n : SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM 7 The form in the special case n = 3 was denoted S 0 in [15] and M in =-=[68]-=-. We give the proof for n = 3; the general case is similar. Rename variables, and let M(x; y; z) = (x 2 + y 2 \Gamma 3z 2 )x 2 y 2 + z 6 = x 4 y 2 + x 2 y 4 + z 6 \Gamma 3x 2 y 2 z 2 : The fact that M... |

24 | A stabilization theorem for Hermitian forms and applications to holomorphic mappings
- D’Angelo, Catlin
- 1996
(Show Context)
Citation Context ...; ffl n xn ): Motzkin and Straus [58] partially generalized P'olya's Theorem to power series in several variables, and discussed some related algebraic questions. Catlin and 16 BRUCE REZNICK D'Angelo =-=[9, 10]-=- have recently generalized P'olya's Theorem (with denominator information) to polynomials in several complex variables. Handelman [32, 33] solved a related question. Suppose a polynomial p in several ... |

23 | Additive number theory: the classical bases, Graduate Texts - Nathanson - 1996 |

21 |
Zur Darstellung definiter Funktionen als Summe von
- Pfister
- 1967
(Show Context)
Citation Context ...(x; y; z) = x 2 y 2 (x 2 + y 2 + z 2 )(x 2 + y 2 \Gamma 2z 2 ) 2 + (x 2 \Gamma y 2 ) 2 z 6 : Hilbert's Theorem was generalized (using entirely different methods) in a celebrated 1967 paper of Pfister =-=[60]-=- (see also [67, x 5]): every p 2 Pn;m is the sum of at most 2 n\Gamma1 squares of rational functions. In 1971, Cassels--Ellison--Pfister [8] proved that M cannot be written as a sum of three squares o... |

17 |
Positive semidefinite biquadratic forms
- Choi
- 1975
(Show Context)
Citation Context ...lder'on [7] had covered some low-dimensional cases and convinced Choi that the result could not be extended. He tried to find the flaw in the proof, and, in doing so, constructed a counterexample (in =-=[11]). He-=- writes [12]: "Without Koga's false proof, I would not have dared construct a counterexample. Actually, I had been haunted by Hilbert's non-constructive example [in [29]] when I was a graduate st... |

16 |
Positivity conditions for bihomogeneous polynomials
- Catlin, D’Angelo
- 1997
(Show Context)
Citation Context ...; ffl n xn ): Motzkin and Straus [58] partially generalized P'olya's Theorem to power series in several variables, and discussed some related algebraic questions. Catlin and 16 BRUCE REZNICK D'Angelo =-=[9, 10]-=- have recently generalized P'olya's Theorem (with denominator information) to polynomials in several complex variables. Handelman [32, 33] solved a related question. Suppose a polynomial p in several ... |

15 |
Valuations and real places in the theory of formally real Geometrie Algebrique Reelle et Formes Quadratiques (Berlin-Heidelberg-New
- Becker
- 1982
(Show Context)
Citation Context ...bout any specific representation of a particular form p 2 Pn;m as a sum of squares of rational functions. Among the many generalizations of the 17th Problem, we mention one in detail. In 1981, Becker =-=[2, 3]-=- gave necessary and sufficient conditions for a rational function p over a formally real field to be a sum of 2k-th powers of rational functions. For such functions over R, the criterion is, roughly s... |

14 |
Extremal positive semidefinite forms
- Choi, Lam
- 1977
(Show Context)
Citation Context ...1 Miraculously, this linear system has rank 27, and the only ternary sextics whose derivatives vanish on Z are the multiples of R. These paragraphs do not completely describe the contents of [14] and =-=[15]-=-; many of the ideas in these papers have yet to be fully developed. My entry into the subject came in late 1976. I was studying the two-dimensional Hamburger moment problem as it applied to an embeddi... |

14 |
Real zeros of positive semidefinite forms
- Choi, Lam, et al.
- 1980
(Show Context)
Citation Context ...hat q is not sos involves the eight zeros of q, as predicted by Hilbert's original argument. 5. Some later developments a. Zeros of psd forms and multiforms. My first collaboration with Choi and Lam, =-=[16]-=-, was largely concerned with the number of zeros of psd forms. Recall that R has the ten zeros of Z. We showed that if p 2 P 3;6 and p has more than ten zeros, viewed projectively, then p is divisible... |

13 |
Even symmetric sextics
- Choi, Lam, et al.
- 1987
(Show Context)
Citation Context ...suggests a study of even symmetric forms. Every psd even symmetric form of degree 2 or 4 is sos, so the simplest "interesting" case is m = 6; psd and sos even symmetric n-ary sextics were an=-=alyzed in [17]-=-. Such a form p can be written p(x 1 ; : : : ; xn ) = ff n X i=1 x 6 i + fi X i6=j x 4 i x 2 j + fl X i!j!k x 2 i x 2 j x 2 k : It is more convenient to express p in terms of a different basis: write ... |

13 |
Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringsches problem
- Hilbert
- 1909
(Show Context)
Citation Context ...\Delta \Delta \Delta + x 2 n ) s = N X k=1sk (ff k1 x 1 + \Delta \Delta \Delta + ff kn xn ) 2s ; where 0 !sk and ff kj 2 R, is a Hilbert Identity. As part of his solution of Waring's Problem, Hilbert =-=[42]-=- proved that Hilbert Identities exist for every n and s, with the additional algebraic property thatsk ; ff kj 2 Q. We shall call these rational Hilbert Identities. There are no known families of expl... |

11 |
On sums of squares and on elliptic curves over functions fields
- Cassels, Ellison, et al.
- 1971
(Show Context)
Citation Context ...olynomial was still not as widely known as it became after O. Taussky-Todd mentioned its existence to A. Pfister who, along with J. W. S. Cassels and W. J. Ellison, did further work in this area [see =-=[8]].&qu-=-ot; Motzkin proved [56, p. 217] that for ns3, (t 2 1 + \Delta \Delta \Delta + t 2 n\Gamma1 \Gamma nu 2 )t 2 1 \Delta \Delta \Delta t 2 n\Gamma1 + u 2n 2 \Delta n;2n : SOME CONCRETE ASPECTS OF HILBERT'... |

10 |
Positive polynomials and product type actions of compact groups
- Handelman
- 1985
(Show Context)
Citation Context ...ated algebraic questions. Catlin and 16 BRUCE REZNICK D'Angelo [9, 10] have recently generalized P'olya's Theorem (with denominator information) to polynomials in several complex variables. Handelman =-=[32, 33]-=- solved a related question. Suppose a polynomial p in several variables has nonnegative coefficients. For which f does there always exist an r so that p r f has nonnegative coefficients? Recently, De ... |

10 | Corrections to An effective version of Pólya’s theorem on positive definite forms
- Loera, Santos
(Show Context)
Citation Context ... question. Suppose a polynomial p in several variables has nonnegative coefficients. For which f does there always exist an r so that p r f has nonnegative coefficients? Recently, De Loera and Santos =-=[54]-=- have turned P'olya's Theorem into an explicit algorithm, and made quantitative estimates for t 0 (f ). The restriction to positive definite forms is necessary. There exist positive semidefinite forms... |

9 |
An example of a positive polynomial, which is not a sum of squares polynomials. A positive but not positive functional
- Schmüdgen
- 1979
(Show Context)
Citation Context ... and not sos. They observe that A is a polynomial in the x i \Gamma x j 's, so it is "really" a form in four variables. (Olympiad contestants were only asked to prove that A is psd!) Konrad =-=Schmudgen [80]-=-, following the program of Gel'fand--Vilenkin, produced a sextic polynomial that homogenizes to a form in \Delta 3;6 : q(x; y; z) = 200(x 3 \Gamma4xz 2 ) 2 +200(y 3 \Gamma4yz 2 ) 2 +(y 2 \Gammax 2 )x(... |

8 |
An Old Question of Hilbert
- CNOI, LAM
- 1977
(Show Context)
Citation Context ...ents, we see that H d (K n ) �� K N . Suppose m is an even integer. A form p 2 Hm (R n ) is called positive semidefinite or psd if p(x 1 ; : : : ; xn )s0 for all (x 1 ; : : : ; xn ) 2 R n . Follow=-=ing [14]-=-, we denote the set of psd forms in Hm (R n ) by Pn;m . Since Pn;m is closed under addition and closed under multiplication by positive scalars, it is a convex cone. In fact, Pn;m is a closed convex c... |

8 |
Über die Darstellung definiter Funktionen durch Quadrate
- Landau
- 1906
(Show Context)
Citation Context ...2 P 3;m\Gamma4 and h 1k 2 Hm\Gamma2 (R 3 ) so that pp 1 = h 2 11 + h 2 12 + h 2 13 . If m = 6 or 8, then p 1 is already known to be the sum of three squares of forms, and hence (as Landau later noted =-=[51]-=-), the four-square identity implies that pp 2 1 = (pp 1 )p 1 is the sum of four squares of forms. If ms10, then the argument can be applied to p 1 , so that there exists p 2 2 P 3;m\Gamma8 with p 1 p ... |

7 |
The real holomorphy ring and sums of 2n-th powers, in Géométrie Algébrique Réelle et Formes
- Becker
- 1982
(Show Context)
Citation Context ...bout any specific representation of a particular form p 2 Pn;m as a sum of squares of rational functions. Among the many generalizations of the 17th Problem, we mention one in detail. In 1981, Becker =-=[2, 3]-=- gave necessary and sufficient conditions for a rational function p over a formally real field to be a sum of 2k-th powers of rational functions. For such functions over R, the criterion is, roughly s... |

7 |
An introduction to real algebra
- Lam
- 1984
(Show Context)
Citation Context ... same degree, then (f=g)(t) can be written as above, but where f jk and g jk are positive definite polynomials of the same degree. There are many expositions of Artin's proof in the literature, e.g., =-=[5, 47, 49, 50]-=-. Ribenboim [74] and Pfister [61] wrote surveys on Hilbert's 17th Problem in the 1970s; two more recent surveys are by Gondard [30] and Scheiderer [78]. The deep connections of Hilbert's 17th Problem ... |

7 |
Some definite polynomials which are not sums of squares of real polynomials
- Robinson
- 1973
(Show Context)
Citation Context ...4], we denote the set of sos forms in Hm (R n ) by \Sigma n;m . It is easy to see that \Sigma n;m is a convex cone; less obvious is the fact that it is closed. This was first proved by R. M. Robinson =-=[77]-=-. Finally, we note the inclusion \Sigma n;m ` Pn;m and define \Delta n;m = Pn;m n \Sigma n;m . If p 2 \Delta n;m , then p may be construed as lying in \Delta n1;m for n 1sn; for even m 1sm, it is easy... |

6 |
Sums of squares in some integral domains
- Choi, Lam, et al.
- 1980
(Show Context)
Citation Context ...on--Pfister [8] proved that M cannot be written as a sum of three squares of rational functions. Their methods were extended to a family of ternary sextics by Christie [21] in 1976. We proved in 1980 =-=[20]-=- that every form that is the sum of two squares of rational functions is in fact the sum of two squares of forms, and asked whether this is true for sums of three squares (p. 254). Very recently, Leep... |

6 |
On sums of squares
- Lax, Lax
- 1978
(Show Context)
Citation Context ...ave worked together (with occasional fourth authors) ever since. d. Examples of Lax--Lax and Schmudgen. Two other forms in \Delta n;m were discovered independently in the 1970's. Anneli and Peter Lax =-=[52]-=- showed that A(x 1 ; x 2 ; x 3 ; x 4 ; x 5 ) := 5 X i=1 Y j 6=i (x i \Gamma x j ); which appeared on the 1971 International Mathematical Olympiad, is psd and not sos. They observe that A is a polynomi... |

6 |
A quantitative version of Hurwitz’ theorem on the arithmetic-geometric inequality
- Reznick
- 1987
(Show Context)
Citation Context ...e Herr Hilbert gezeigt hat, positive Formen, welche nicht als Summen von Formenquadraten darstellbar sind." The Hurwitz construction, which can also be found in [35, p. 55], is simplified somewha=-=t in [69]-=-. In 1893, Hilbert [39] generalized his earlier result on P 3;4 ; his proof seems to be non-constructive, and lacks a modern exposition. Suppose p 2 P 3;m with ms6. Then there exist p 1 2 P 3;m\Gamma4... |

5 |
A constructible continuous solution to Hilbert’s 17th problem, and other results in real Algebraic
- Delzell
- 1980
(Show Context)
Citation Context ...sitive definite form can have a bad point. Bad points were first noted by E. G. Straus in an unpublished 1956 letter to G. Kreisel. An extensive history of this topic can be found in Delzell's thesis =-=[22]-=-, and in his [25, 26]. An example from [22] is D[w; x; y; z] := w 2 S(x; y; z) + z 8 , where S 2 \Delta 3;6 is defined in x4c. It is easy to show that D has a bad point at (w; x; y; z) = (1; 0; 0; 0).... |

5 |
Case distinctions are necessary for representing polynomials as sums of squares
- Delzell
- 1981
(Show Context)
Citation Context ...ional questions in geometry and the study of totally positive elements in number fields. Modern discussions of the geometric roots of Hilbert's 17th Problem have been made by Prestel [66] and Delzell =-=[23]-=-. The Hilbert-Landau-Siegel Theorem states that if x is an element in a number field F that is positive in each embedding of F into R, then x is a sum of four squares. Elements that are not totally po... |

5 |
A continuous and rational solution to Hilbert’s 17th problem and several
- Delzell, Gonz'alez-Vega, et al.
- 1993
(Show Context)
Citation Context ...resk (p)s0 is linear in p. There has been considerable interest in the representations of p as a sum of squares of rational functions with continuous dependence on p, such as the one given above; see =-=[24, 27]-=-. Such a formula cannot hold over all of Pn;m . It is not hard to show that if p 2 Pn;m is not positive definite, then pG r n cannot be written as a sum of (2r + m)-th powers of linear forms over R. T... |

5 |
Über die Zerlegung strikte definiter Formen in Quadrate
- Habicht
- 1940
(Show Context)
Citation Context ...ich bemerkt werden, dass die Darstellung einigermassen in Zusammenhang mit einer Fragestellung von Hilbert steht, die kurzlich durch E. Artin mit tiefgehenden Mittleln gelost wurde." In 1940, Hab=-=icht [31]-=- (see also [35, pp. 300--304]) used P'olya's Theorem to prove directly that a (not necessarily even) positive definite form is a sum of squares of rational functions. The denominators in the represent... |

5 |
Hilbert’s seventeenth problem and related problems on definite forms
- Pfister
- 1974
(Show Context)
Citation Context ...s above, but where f jk and g jk are positive definite polynomials of the same degree. There are many expositions of Artin's proof in the literature, e.g., [5, 47, 49, 50]. Ribenboim [74] and Pfister =-=[61]-=- wrote surveys on Hilbert's 17th Problem in the 1970s; two more recent surveys are by Gondard [30] and Scheiderer [78]. The deep connections of Hilbert's 17th Problem with logic were initiated by A. R... |

5 |
Hilbert’s theorem on positive ternary quartics, Contemp
- Swan
(Show Context)
Citation Context ... three squares of quadratic forms. (For an elementary proof, with "three" replaced by "five", see [15]; modern expositions of Hilbert's proof have been given by Cassels (in [67, pp=-=. 89--93]) and Swan [82]-=-.) Hilbert's second result is that the preceding are the only cases for which \Delta n;m = ;. That is, if ns3 and ms6 or ns4 and ms4, then there exist forms p 2 Pn;m that are not sos. These can be der... |

4 |
Sums of powers in rings and the real holomorphy
- Becker, Powers
- 1996
(Show Context)
Citation Context ...at the end of x7; however the coefficients and polynomials have real coefficients, rather than the rational ones requested by Becker. One can deduce from recent work of 6 BRUCE REZNICK Becker--Powers =-=[4]-=- that there is a representation of B(t) as a sum of 2k-th powers in which each g jk is positive definite. Schmid has also recently shown [79] that if f and g are positive definite polynomials in one v... |

4 |
Deciding eventual positivity of polynomials, Ergodic Theory Dynamical Systems 6
- Handelman
- 1986
(Show Context)
Citation Context ...ated algebraic questions. Catlin and 16 BRUCE REZNICK D'Angelo [9, 10] have recently generalized P'olya's Theorem (with denominator information) to polynomials in several complex variables. Handelman =-=[32, 33]-=- solved a related question. Suppose a polynomial p in several variables has nonnegative coefficients. For which f does there always exist an r so that p r f has nonnegative coefficients? Recently, De ... |

4 |
Über ternäre definite Formen, Acta Math. 17 (1893) 169–197; see Ges
- Hilbert
- 1933
(Show Context)
Citation Context ...hat, positive Formen, welche nicht als Summen von Formenquadraten darstellbar sind." The Hurwitz construction, which can also be found in [35, p. 55], is simplified somewhat in [69]. In 1893, Hil=-=bert [39]-=- generalized his earlier result on P 3;4 ; his proof seems to be non-constructive, and lacks a modern exposition. Suppose p 2 P 3;m with ms6. Then there exist p 1 2 P 3;m\Gamma4 and h 1k 2 Hm\Gamma2 (... |

4 | Synthesis of finite passive n-ports with prescribed positive real matrices of several variables - Koga - 1968 |

4 |
On ordered fields and definite forms
- Robinson
- 1955
(Show Context)
Citation Context ...urveys on Hilbert's 17th Problem in the 1970s; two more recent surveys are by Gondard [30] and Scheiderer [78]. The deep connections of Hilbert's 17th Problem with logic were initiated by A. Robinson =-=[75, 76]-=- in the mid-1950's; Delzell has written [25] a recent detailed history of logicians' interest in Hilbert's 17th Problem. The spectacular development of real algebra and real algebraic geometry is well... |

4 |
Integral solution of Hilbert’s seventeenth problem
- Stengle
(Show Context)
Citation Context .... If H 2 \Delta 5;4 denotes the Horn form (see x4c), then it can be shown that for every rs1, H(x r 1 ; : : : ; x r 5 ) is not sos. A related question involves taking odd powers of psd forms. Stengle =-=[81]-=- proved in 1979 that for ms1, every odd power of x 2m+1 1 x 2m+1 3 + (x 2 2 x 2m\Gamma1 3 \Gamma x 2m+1 1 \Gamma x 1 x 2m 3 ) 14 BRUCE REZNICK is psd and not sos. It can be shown that this property al... |

3 |
A note on biquadratic forms
- Calderón
- 1973
(Show Context)
Citation Context ...adratic forms in two different sets of variables. Choi learned of a paper [48, p. 14] by an electrical engineer, purporting to show that every psd biquadratic form is sos. A recent paper of Calder'on =-=[7]-=- had covered some low-dimensional cases and convinced Choi that the result could not be extended. He tried to find the flaw in the proof, and, in doing so, constructed a counterexample (in [11]). He w... |

3 |
Positive Definite Rational Functions of Two Variables Which Are Not the Sum of Three Squares
- Christie
- 1976
(Show Context)
Citation Context ...nctions. In 1971, Cassels--Ellison--Pfister [8] proved that M cannot be written as a sum of three squares of rational functions. Their methods were extended to a family of ternary sextics by Christie =-=[21]-=- in 1976. We proved in 1980 [20] that every form that is the sum of two squares of rational functions is in fact the sum of two squares of forms, and asked whether this is true for sums of three squar... |

3 |
On a theorem in differentiation, and its application to spherical harmonics
- Hobson
- 1892
(Show Context)
Citation Context ...ble function of one variable, then h(D)F (Gn ) = X k0 2 d 2 2k k! \Delta k (h)F (d\Gammak) (Gn ): The right-hand side is a finite sum; if k ? d=2, then \Delta k (h) = 0. Hobson's Theorem is proved in =-=[43, 44], see also-=- [45]; it was lauded by Hardy [34] as an "elegant theorem in formal differentiation". Now, set F (t) = t s , so F (j) (t) = (s) j t s\Gammaj : h(D)G s n = X k0 (s) d\Gammak 2 2k\Gammad k! \D... |

3 |
The theory of ordered fields, Ring theory and algebra
- Lam
- 1979
(Show Context)
Citation Context |

3 |
Forms derived from the arithmetic-geometric inequality
- Reznick
- 1989
(Show Context)
Citation Context ... This implies the extremality of the forms M (with a = (2; 1; 0), b = (1; 2; 0), c = (0; 0; 3), d = (1; 1; 1)) and S (with a = (2; 1; 0), b = (0; 2; 1), c = (1; 0; 2), d = (1; 1; 1)). An agiform (see =-=[70]-=-) is a form derived by making even monomial substitutions into the arithmetic-geometric inequality. Suppose a i 2 (2Z) n + andsi are positive reals, P isi = 1, so that P isi a i = b 2 Z n . Then the a... |

3 |
Further remarks on ordered fields and definite forms
- Robinson
- 1956
(Show Context)
Citation Context ...urveys on Hilbert's 17th Problem in the 1970s; two more recent surveys are by Gondard [30] and Scheiderer [78]. The deep connections of Hilbert's 17th Problem with logic were initiated by A. Robinson =-=[75, 76]-=- in the mid-1950's; Delzell has written [25] a recent detailed history of logicians' interest in Hilbert's 17th Problem. The spectacular development of real algebra and real algebraic geometry is well... |

2 |
Positive sextics and Schur’s inequalities
- Choi, Lam, et al.
- 1991
(Show Context)
Citation Context ...Gamma u)(v \Gamma w) + w r (w \Gamma u)(w \Gamma v)s0 if r; u; v; ws0: (Take r = 1 and (u; v; w) = (x 2 ; y 2 ; z 2 ) to obtain R.) For much more on Schur's inequalities and related sextic forms, see =-=[18]-=-. It is easy to see that R = 0 on the set Z := f(1; \Sigma1; \Sigma1); (1; \Sigma1; 0); (1; 0; \Sigma1); (0; 1; \Sigma1)g: If R = P k h 2 k , where each h k is a ternary cubic, then h k vanishes on Z.... |