Formulation of the . . . Definitions First Result Need for a General Case If a Polynomial Mapping Is Definitions Rectifiable, then the Second Result Discussion Rectifying Polynomial Tarski-Seidenberg . . . Proof of First Result Automorphism Can Be Home Page Algorithmically Computed Title Page ◭◭ ◮◮ Julio Urenda 1 , 2 , David Finston 1 , and Vladik Kreinovich 3 ◭ ◮ 1 Department of Mathematical Sciences New Mexico State University, Las Cruces, NM 88003, USA Page 1 of 19 jcurenda@utep.edu, dfinston@nmsu.edu 2 Department of Mathematical Sciences Go Back 3 Department of Computer Science Full Screen University of Texas at El Paso El Paso, TX 79968, USA, vladik@utep.edu Close Quit
Formulation of the . . . Definitions 1. Formulation of the Problem First Result • Let C denote the field of all complex numbers. Need for a General Case • A polynomial mapping α : C n → C n is called a poly- Definitions Second Result nomial automorphism if: Discussion – this mapping a bijection, and Tarski-Seidenberg . . . – the inverse mapping β = α − 1 is also polynomial. Proof of First Result • A polynomial mapping ϕ : C k → C n is called rectifi- Home Page able if: Title Page – these exists a polynomial automorphism ◭◭ ◮◮ α : C n → C n ◭ ◮ – for which α ( ϕ ( t 1 , . . . , t k )) = ( t 1 , . . . , t k , 0 , . . . ) for all Page 2 of 19 ( t 1 , . . . , t k ). Go Back • Most existing proofs of rectifiability just prove the ex- istence of a rectifying automorphism α . Full Screen • In this talk, we show how to compute α . Close Quit
Formulation of the . . . Definitions 2. Definitions First Result • We will formulate two versions of the main result: Need for a General Case Definitions – for the case when the coefficients of the original Second Result polynomial mapping are algebraic numbers, and Discussion – for the general case, when these coefficients are not Tarski-Seidenberg . . . necessarily algebraic. Proof of First Result • A number is called algebraic if this number is a root of Home Page a non-zero polynomial with integer coefficients. Title Page • In the computer, an algebraic real number can be rep- ◭◭ ◮◮ resented by: ◭ ◮ – the integer coefficients of the corresponding poly- nomial and Page 3 of 19 – by a rational-valued interval that contains only this Go Back root. Full Screen • Once this information is given, we can compute the Close corresponding root with any given accuracy. Quit
Formulation of the . . . Definitions 3. First Result First Result • Lemma. Need for a General Case Definitions – If a polynomial mapping ϕ with algebraic coeffi- Second Result cients is rectifiable, Discussion – then there exists a rectifying polynomial automor- Tarski-Seidenberg . . . phism α with algebraic coefficients. Proof of First Result • Proposition. There exists an algorithm that: Home Page – given a rectifiable polynomial mapping ϕ with alge- Title Page braic coefficients, ◭◭ ◮◮ – computes the coefficients of a polynomial automor- ◭ ◮ phism α that rectifies ϕ . Page 4 of 19 Go Back Full Screen Close Quit
Formulation of the . . . Definitions 4. Need for a General Case First Result • In general, the coefficients of the original mapping ϕ Need for a General Case are not necessarily algebraic. Definitions Second Result • These coefficients may not even be computable. Discussion • It is desirable to extend this algorithm to this general Tarski-Seidenberg . . . case. Proof of First Result Home Page • When the coefficients are not necessarily computable, we cannot represent them in a computer. Title Page • So, we need to extend the usual notion of an algorithm ◭◭ ◮◮ to cover this case. ◭ ◮ Page 5 of 19 Go Back Full Screen Close Quit
Formulation of the . . . Definitions 5. Definitions First Result • By a generalized algorithm , we mean a sequence of the Need for a General Case following elementary operations with real numbers: Definitions Second Result – adding, subtracting, multiplying, and dividing Discussion numbers; Tarski-Seidenberg . . . – checking whether a number is equal to 0, whether Proof of First Result it is positive, and whether it is negative; Home Page – given the coefficients of a polynomial that has a Title Page root, returning one of the roots. ◭◭ ◮◮ • Of course, when the real numbers are algebraic, these ◭ ◮ operations are algorithmically computable. Page 6 of 19 Go Back Full Screen Close Quit
Formulation of the . . . Definitions 6. Second Result First Result • Proposition. There exists a generalized algorithm Need for a General Case that: Definitions Second Result – given the coefficients of a rectifiable polynomial Discussion mapping ϕ , Tarski-Seidenberg . . . – computes the coefficients of a polynomial automor- Proof of First Result phism α that rectifies ϕ . Home Page • This shows that: Title Page – if a polynomial mapping is rectifiable, ◭◭ ◮◮ – then the corresponding rectification can be algo- ◭ ◮ rithmically computed. Page 7 of 19 Go Back Full Screen Close Quit
Formulation of the . . . Definitions 7. Discussion First Result • Our proof uses the Tarski algorithm. Need for a General Case Definitions • As the length ℓ of the formula increases, the running Second Result time of this algorithm grows faster than 2 2 ℓ . Discussion • Thus, from the application viewpoint, it is desirable to Tarski-Seidenberg . . . come up with a faster algorithm. Proof of First Result Home Page • For some important cases, such faster algorithms were also proposed. Title Page • These faster algorithms can be applied to other fields ◭◭ ◮◮ (and rings) as well. ◭ ◮ • They are described in J. Urenda’s NMSU PhD disser- Page 8 of 19 tation Algorithmic Aspects of the Embedding Problem . Go Back Full Screen Close Quit
Formulation of the . . . Definitions 8. Tarski-Seidenberg Algorithm: Reminder First Result • This algorithm deals with the first-order theory of real Need for a General Case numbers: follows: Definitions Second Result – we start with real-valued variables x 1 , . . . , x n ; Discussion – elementary formulas : P = 0, P > 0, or P ≥ 0, Tarski-Seidenberg . . . where P is a polynomial with integer coefficients; Proof of First Result – a general formula can be obtained from elementary Home Page formulas by using: Title Page • logical connectives (“and” &, “or” ∨ , “implies” ◭◭ ◮◮ → , and “not” ¬ ) and • quantifiers over real numbers ( ∀ x i and ∃ x i ). ◭ ◮ Page 9 of 19 • Example: every quadratic polynomial with non- negative determinant has a solution: Go Back ∀ a ∀ b ∀ c (( b 2 − 4 a · c ≥ 0) → ∃ x ( a · x 2 + b · x + c = 0)) . Full Screen Close Quit
Formulation of the . . . Definitions 9. Tarski-Seidenberg Algorithm (cont-d) First Result • Tarski designed an algorithm that: Need for a General Case Definitions – given a formula from this theory, Second Result – returns 0 or 1 depending on whether this formula Discussion is true or not. Tarski-Seidenberg . . . • Seidenberg used a similar construction to Proof of First Result Home Page – reduce each first-order formula with free variables Title Page – to a quantifier-free form. ◭◭ ◮◮ • It follows that if a formula with free variables has a solution, then it also has an algebraic solution. ◭ ◮ Page 10 of 19 Go Back Full Screen Close Quit
Formulation of the . . . Definitions 10. Proof of Lemma First Result • Let d be the largest degree of polynomials α i and β i Need for a General Case forming the mappings α and β = α − 1 . Definitions Second Result • Each of these polynomial can be described by listing Discussion real and imaginary values of all its coefficients. Tarski-Seidenberg . . . • The condition that α and β are inverse to each other Proof of First Result means that ∀ z 1 . . . ∀ z n (& i α i ( β ( z 1 , . . . , z n )) = z i ) and Home Page ∀ z 1 . . . ∀ z n (& j β j ( α ( z 1 , . . . , z n )) = z j ) . Title Page ◭◭ ◮◮ • Substituting the expressions for α and β in terms of their coefficients, we get a first order formula. ◭ ◮ • Similarly, the condition that α rectifies ϕ is Page 11 of 19 Go Back ∀ t 1 . . . ∀ t k (& ℓ α ℓ ( ϕ ( t 1 , . . . , t k ) = t ℓ ) – a first-order formula . Full Screen • Thus, if ∃ a solution, then ∃ a solution in which all coefficients of all polynomials α i and β i are algebraic. Close Quit
Formulation of the . . . Definitions 11. Proof of First Result First Result • Due to Tarski’s algorithm: Need for a General Case Definitions – for each tuple of algebraic numbers, Second Result – we can check whether the corresponding polynomi- Discussion als constitute a rectifying automorphism. Tarski-Seidenberg . . . • To find the desired polynomial mappings α and β with Proof of First Result algebraic coefficients, it is sufficient to: Home Page – enumerate all possible tuples of such coefficients, Title Page – try them one by one, ◭◭ ◮◮ – until we find a tuple which corresponds to the rec- ◭ ◮ tifying automorphism. Page 12 of 19 • Since we assumed that a rectification is possible, we Go Back will eventually find the desired coefficient. Full Screen • The only thing that needs to be clarified is how to enumerate all possible tuples of algebraic numbers. Close Quit
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