## Addition Theorems (1965)

Citations: | 8 - 0 self |

### BibTeX

@MISC{A65additiontheorems,

author = {Ehud Hrushovski A and Itamar Pitowsky B},

title = {Addition Theorems},

year = {1965}

}

### OpenURL

### Abstract

theorem and the effectiveness of Gleason’s

### Citations

273 |
Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
- Cox, Little, et al.
- 1992
(Show Context)
Citation Context ...with Q ⊂ K ⊂ Q and a unit u ∈ K(x) such that g(u −1 ), u(h) ∈ K(x) and [K : Q] ≤ 3. The proof is a straightforward application of Gröbner bases and the well– known Extension Theorem, see for instance =-=[3]-=-. 3 Fixing group and fixed field In this section we introduce several simple notions from classical Galois theory. Let Γ(K) = AutKK(x) (we will write simply Γ if there can be no confusion about the fi... |

248 |
The problem of hidden variables in quantum mechanics
- Kochen, Specker
- 1975
(Show Context)
Citation Context ...mple version of Gleason’s theorem. It states that there does not exist a bi-valued probability function on the rays (onedimensional subspaces) of a Hilbert space of dimension X3: A later improvement (=-=Kochen & Specker, 1967-=-) showed that there is a finite subset of rays on which no bivalued probability function exists. A natural question to ask is why KS is considered an improvement. If, for example, contextuality follow... |

224 | Mathematical Logic - Shoenfield - 1967 |

171 |
Measures on the closed subspaces of a hilbert space
- Gleason
- 1957
(Show Context)
Citation Context ... which every quantum state can be approximated. r 2003 Elsevier Ltd. All rights reserved. Keywords: Gleason’s theorem; Logical compactness; Constructive mathematics 1. Introduction Gleason’s theorem (=-=Gleason, 1957-=-) plays an important role in the foundations of quantum mechanics. On the positive side it demonstrates how the probabilistic structure of quantum theory follows from its logical structure, that is, t... |

167 | On the problem of hidden variables in quantum mechanics - Bell - 1966 |

163 |
Modern Computer Algebra
- Gathen, Gerhard
- 2002
(Show Context)
Citation Context ...olynomial in the given field. We analyze the bit complexity when the ground field is the rational number Q. We will use several well–known results about complexity, those can be consulted in the book =-=[7]-=-. In the following, M denotes a multiplication time, so that the product of two polynomials in K[x] with degree at most m can be computed with at most M(m) arithmetic operations. If K supports the Fas... |

159 | Foundations of Quantum Physics - Piron - 1976 |

95 |
Polynomials with Special Regard to Reducibility
- Schinzel
- 2000
(Show Context)
Citation Context ...f (j) ij ◦ f(j) ij+1 and f (j+1) ij ◦ f (j+1) ij+1 ◦ f(j) ij+1 = f (j+1) ij ◦ f (j+1) ij+1 are equivalent. is a bidecomposition. PROOF. See [13] for K = C, [5] for characteristic zero fields and [6], =-=[15]-=- for the general case. ✷ Unlike for polynomials, it is not true that all complete decompositions of a rational function have the same length, as shown in Example 20. The paper [10] presents a detailed... |

85 |
der Waerden, “Modern Algebra
- van
- 1949
(Show Context)
Citation Context ...|H|. Let H = {h1 = x, . . .,hm}. Let m∏ P(T) = (T − hi) ∈ K(x)[T]. i=1 We will see that P(T) is the minimum polynomial of x over Fix(H) ⊂ K(x). A classical proof of Lüroth’s Theorem (see for instance =-=[17]-=-) states that any non–constant coefficient of the minimum polynomial generates Fix(H), and we are done. It is obvious that P(x) = 0, as x is always in H. It is also clear that P(T) ∈ Fix(H)[T], as its... |

72 |
Prime and composite polynomials
- Ritt
- 1922
(Show Context)
Citation Context ...x − 1 . In the next section we will use these tools to investigate the number of components of a rational function. 4 Ritt’s Theorem and number of components One of the classical Ritt’s Theorems (see =-=[13]-=-) describes the relation among the different decomposition chains of a tame polynomial. Essentially, all the decompositions have the same length and are related in a rather simple way. Definition 23 A... |

53 |
Lectures on the icosahedron and the solutions of equations of the fifth degree
- Klein
- 1956
(Show Context)
Citation Context ...and fields in general, as the previous example shows for Q. On the other hand, only some finite groups can be subgroups of PGL2(K). The only finite subgroups of PGL2(C) are Cn, Dn, A4, S4 and A5, see =-=[11]-=-. In fact, this is true for any algebraically closed field of characteristic zero (it suffices that it contains all roots of unity). Among these groups, only A4 has subgroup chains of different length... |

40 |
Permutable rational functions
- Ritt
- 1923
(Show Context)
Citation Context ...l with coefficients over a finite field. The problem for rational functions is strongly related to the open problem of the classes of rational functions which commute with respect to composition, see =-=[14]-=-. In this section we will give some ideas about the relation between complete decompositions and subgroup chains that appear by means of Galois Theory. Now we present another degree 12 function, this ... |

34 | New results on quantifier elimination over real closed fields and applications to constraint databases - Basu - 1999 |

33 |
A rational function decomposition algorithm by near-separated polynomials
- Alonso, Gutierrez, et al.
- 1995
(Show Context)
Citation Context ...ion states that if a polynomial is indecomposable in a certain coefficient field, then it is also indecomposable in any extension of that field. This is also false for rational functions, see [4] and =-=[1]-=-. We look for bounds for the degree of the extension in which we need to take the coefficients if a rational function with coefficients in Q has a decomposition in a larger field. In this paper we pre... |

33 |
On the invariance of chains of fields
- Fried, MacRae
- 1969
(Show Context)
Citation Context ...(ii) f (j) ij ◦ f(j) ij+1 and f (j+1) ij ◦ f (j+1) ij+1 ◦ f(j) ij+1 = f (j+1) ij ◦ f (j+1) ij+1 are equivalent. is a bidecomposition. PROOF. See [13] for K = C, [5] for characteristic zero fields and =-=[6]-=-, [15] for the general case. ✷ Unlike for polynomials, it is not true that all complete decompositions of a rational function have the same length, as shown in Example 20. The paper [10] presents a de... |

32 | Finite precision measurement nullifies the KochenSpecker theorem” Phys
- Meyer
- 1999
(Show Context)
Citation Context ...n states than continuity. Conceptually, one of the important outcomes of Gleason’s theorem is the 1 This result has been used in Breuer (2002) to give an argument against the ‘‘nullification’’ of KS (=-=Meyer, 1999-=-; Clifton & Kent, 2000; see also, Pitowsky, 1983, 1985; Appleby, 2002).ARTICLE IN PRESS 182 E. Hrushovski, I. Pitowsky / Stud. in Hist. and Phil. of Mod. Phys. 35 (2004) 177–194 indeterminacy princip... |

32 | Rational function decomposition
- Zippel
- 1991
(Show Context)
Citation Context ...ength; as far as we know this is the first example in Q of this kind. 2 Extension of the coefficient field Several algorithms for decomposing univariate rational functions are known, see for instance =-=[18]-=- and [1]. In all cases, the complexity of the algorithm grows enormously when the coefficient field is extended. A natural question about decomposition is whether it depends on the coefficient field, ... |

26 | Simulating quantum mechanics by noncontextual hidden variables
- Clifton, Kent
(Show Context)
Citation Context ... continuity. Conceptually, one of the important outcomes of Gleason’s theorem is the 1 This result has been used in Breuer (2002) to give an argument against the ‘‘nullification’’ of KS (Meyer, 1999; =-=Clifton & Kent, 2000-=-; see also, Pitowsky, 1983, 1985; Appleby, 2002).ARTICLE IN PRESS 182 E. Hrushovski, I. Pitowsky / Stud. in Hist. and Phil. of Mod. Phys. 35 (2004) 177–194 indeterminacy principle. Casting it in our ... |

19 |
Computation of the Galois groups of the resolvent factors for the direct and inverse Galois problems. In Applied algebra, algebraic algorithms and error-correcting codes
- Valibouze
- 1995
(Show Context)
Citation Context ...inite then Γ(K) is finite too, an the proof of Theorem 15 provides a non–constant rational function that generates Fix(Γ(K)). Algorithms for computing several aspects of Galois theory can be found in =-=[16]-=-. Unfortunately, it is not true in general that [K(x) : K(f)] = |G(f)|; there 7is no bijection between intermediate fields and subgroups of the fixing group of a given function. Anyway, we can obtain... |

15 | An elementary proof of Gleason's theorem - COOKE, |MORAN - 1985 |

14 | Betting on the Outcomes of Measurements: A Bayesian Theory of Quantum
- Pitowsky
(Show Context)
Citation Context ...We do not present arguments in favor or against any particular interpretation of quantum mechanics. However, it is not difficult to trace the quantum logical motivation behind the results. Elsewhere (=-=Pitowsky, 2003-=-) it is argued that quantum probability can be seen as Bayesian betting ratios in finite gambles provided that we take the logical relations among the observables seriously. Here we complete this pict... |

13 | A constructive proof of Gleason’s theorem
- Richman, Bridges
- 1999
(Show Context)
Citation Context ...explicitly show where the minimum is obtained. However, Billinge (1997) pointed out that there is an alternative constructive formulation of Gleason’s theorem, and a constructive proof soon followed (=-=Richman & Bridges, 1999-=-). Apart from its intrinsic value this issue should be of particular interest to those who hold that quantum mechanics is a theory of information. Since all the information that we shall ever possess ... |

10 | Getting contextual and nonlocal elements{of{reality the easy - CLIFTON - 1993 |

10 | Unirational fields of transcendence degree one and functional decomposition
- Gutierrez, Rubio, et al.
- 2001
(Show Context)
Citation Context ...e sense explained above, all the decompositions of f. One way to find a bound is by means of a result that relates decomposition and factorization. We state the main definition and theorems here, see =-=[9]-=- for proofs and other details. Definition 7 A rational function f ∈ K(x) is in normal form if deg fN > deg fD and fN(0) = 0 (thus fD(0) = 0). Theorem 8 (i) Given f ∈ K(x), if deg f < |K| then there e... |

8 |
Deterministic model of spin and statistics
- Pitowsky
- 1983
(Show Context)
Citation Context ...f the important outcomes of Gleason’s theorem is the 1 This result has been used in Breuer (2002) to give an argument against the ‘‘nullification’’ of KS (Meyer, 1999; Clifton & Kent, 2000; see also, =-=Pitowsky, 1983-=-, 1985; Appleby, 2002).ARTICLE IN PRESS 182 E. Hrushovski, I. Pitowsky / Stud. in Hist. and Phil. of Mod. Phys. 35 (2004) 177–194 indeterminacy principle. Casting it in our language it says that any ... |

8 |
Infinite and finite Gleason’s theorems and the logic of indeterminacy
- Pitowsky
- 1998
(Show Context)
Citation Context ...’s theorem. More precisely, the fact that there is a finite set of rays on which no bi-valued probability function exists follows from the application of the compactness theorem to Gleason’s theorem (=-=Pitowsky, 1998-=-; a similar point is made in Bell, 1996). Of course, KS provided an explicit construction of the set and not just an abstract proof of its existence. However, the chain of reasoning which begins with ... |

7 |
Gleason’s theorem is not constructively provable
- Hellman
- 1993
(Show Context)
Citation Context ...iveness of Gleason’s theorem Recently, there has been an interesting discussion on the question whether Gleason’s theorem has a proof which is acceptable by the standards of constructive mathematics (=-=Hellman, 1993-=-; Billinge, 1997). The discussion culminated in Richman and Bridges (1999), who gave a constructive formulation and proof of the theorem. Our aim is to give a (much shorter) proof of the conditional s... |

7 |
A polynomial decomposition algorithm over factorial domains
- Gutierrez
- 1991
(Show Context)
Citation Context ...ome light on the rational case. Definition 3 f ∈ K[x] is tame when char K does not divide deg f. The next theorem shows that tame polynomials behave well under extension of the coefficient field, see =-=[8]-=-. It is based on the concept of approximate root of a polynomial, which always exists for tame polynomials, and is also the key to some other structural results in the tame polynomial case. Theorem 4 ... |

5 | Quantum mechanics and value definiteness - Pitowsky - 1985 |

5 |
Cardinal conditions for strong Fubini theorems
- Shipman
- 1990
(Show Context)
Citation Context ...at the boundedness of frame functions might be automatic. Coming back to the remark at the end of Section 2, we also note that in such models there are no large OCS 2 on which bi-valued states exist (=-=Shipman, 1990-=-). It seems instructive at all events to compare Lemma 3 to what one obtains when Y satisfies the stronger assumption, that every function (bounded or otherwise) vanishes. Then one can replace in Lemm... |

5 | On Ritt’s decomposition theorem in the case of finite fields, Finite Fields
- Gutierrez, Sevilla
(Show Context)
Citation Context ...ero fields and [6], [15] for the general case. ✷ Unlike for polynomials, it is not true that all complete decompositions of a rational function have the same length, as shown in Example 20. The paper =-=[10]-=- presents a detailed study of this problem for non tame polynomial with coefficients over a finite field. The problem for rational functions is strongly related to the open problem of the classes of r... |

3 |
Existential contextuality and the models of
- Appleby
- 2002
(Show Context)
Citation Context ...es of Gleason’s theorem is the 1 This result has been used in Breuer (2002) to give an argument against the ‘‘nullification’’ of KS (Meyer, 1999; Clifton & Kent, 2000; see also, Pitowsky, 1983, 1985; =-=Appleby, 2002-=-).ARTICLE IN PRESS 182 E. Hrushovski, I. Pitowsky / Stud. in Hist. and Phil. of Mod. Phys. 35 (2004) 177–194 indeterminacy principle. Casting it in our language it says that any two nonorthogonal, no... |

3 |
A constructive formulation of Gleason’s theorem
- Billinge
- 1997
(Show Context)
Citation Context ...son’s theorem Recently, there has been an interesting discussion on the question whether Gleason’s theorem has a proof which is acceptable by the standards of constructive mathematics (Hellman, 1993; =-=Billinge, 1997-=-). The discussion culminated in Richman and Bridges (1999), who gave a constructive formulation and proof of the theorem. Our aim is to give a (much shorter) proof of the conditional statement: If Gle... |

3 |
Application of univariate rational decomposition to Monstrous Moonshine (in Spanish
- McKay, Sevilla
(Show Context)
Citation Context ...function arises in the context of Monstrous Moonshine as a rational relationship between two modular functions (see for example the classical [2] for an overview of this broad topic, or the reference =-=[12]-=-, in Spanish, for the computations in 14which this function appears). Example 25 Let f ∈ Q(x) be the following degree 12 function: f = x3 (x + 6) 3 (x 2 − 6 x + 36) 3 (x − 3) 3 (x 2 + 3 x + 9) 3 . f ... |

2 |
Logical reflections on the Kochen–Specker Theorem
- Bell
- 1996
(Show Context)
Citation Context ...here is a finite set of rays on which no bi-valued probability function exists follows from the application of the compactness theorem to Gleason’s theorem (Pitowsky, 1998; a similar point is made in =-=Bell, 1996-=-). Of course, KS provided an explicit construction of the set and not just an abstract proof of its existence. However, the chain of reasoning which begins with the compactness theorem has the advanta... |

2 |
The Kochen and Specker Theorem” Stanford Encyclopedia of Philosophy http://plato.stanford.edu
- Held
- 2000
(Show Context)
Citation Context ...’s theorem and logical compactness The Kochen and Specker’s theorem (Kochen & Specker, 1967) is often regarded as an improvement on the infinite argument based on Gleason’s theorem (see, for example, =-=Held, 2000-=-). It is less often noted that KS can actually be derived from Gleason’s theorem. More precisely, the fact that there is a finite set of rays on which no bi-valued probability function exists follows ... |

1 | Another no-go theorem for hidden variable models of inaccurate spin measurement - Breuer - 2002 |

1 | Substitution and truth in quantum logic - Pitowsky - 1982 |