The Model Companion of Differential Fields With Free Operators

Sánchez, Omar León; Moosa, Rahim

doi:10.1017/jsl.2015.76

Cited by 4 publications

(3 citation statements)

References 9 publications

(38 reference statements)

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“…is injective as a composition of two injective homomorphisms. Hence, the restriction 16 Let (X, π 1 , π 2 ) be a triple with X bounded by n. Then, for every , the number of maximal trains of length in X does not exceed Components(m, n).…”

Section: Algorithms Yield Upper Bounds In Differential Algebramentioning

confidence: 99%

“…We believe that our method can also be applied to obtain bounds for other algorithms in differential algebra, such as [1, Algorithm 3.6], and for algorithms from other theories, e.g., [7, Algorithm 3] for systems of difference equations. Because the reducibility of a polynomial can be expressed as a first-order existential formula, it seems plausible that the same methods could be applied to other algorithms dealing with difference [5] and differential-difference [6] equations that use factorization, because the corresponding theories satisfy the requirements of our approach [14, 16, 23]. However, we leave these for future research.…”

Section: Introductionmentioning

confidence: 99%

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Algorithms yield upper bounds in differential algebra

Li¹,

Ovchinnikov

Pogudin

et al. 2021

Can. J. Math.-J. Can. Math.

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Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields).We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

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Section: Algorithms Yield Upper Bounds In Differential Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algorithms yield upper bounds in differential algebra

Li¹,

Ovchinnikov

Pogudin

et al. 2021

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

show abstract

“…[7,Algorithm 3] for systems of difference equations. Since the reducibility of a polynomial can be expressed as a first-order existential formula, it seems plausible that the same methods could be applied to other algorithms dealing with difference [5] and differential-difference [6] equations that use factorization because the corresponding theories satisfy the requirements of our approach [14,17,23]. However, we leave these for future research.…”

Section: Introductionmentioning

confidence: 99%

Algorithms yield upper bounds in differential algebra

Li¹,

Ovchinnikov²,

Pogudin³

et al. 2020

Preprint

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Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields).We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

show abstract

Differential Weil descent

Sánchez

Tressl

2021

Communications in Algebra

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The Model Companion of Differential Fields With Free Operators

Cited by 4 publications

References 9 publications

Algorithms yield upper bounds in differential algebra

Algorithms yield upper bounds in differential algebra

Algorithms yield upper bounds in differential algebra

Differential Weil descent

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