13 research outputs found
Effective Differential Nullstellensatz for Ordinary DAE Systems with Constant Coefficients
We give upper bounds for the differential Nullstellensatz in the case of
ordinary systems of differential algebraic equations over any field of
constants of characteristic . Let be a set of differential
variables, a finite family of differential polynomials in the ring
and another polynomial which vanishes at
every solution of the differential equation system in any
differentially closed field containing . Let and . We
show that belongs to the algebraic ideal generated by the successive
derivatives of of order at most , for a suitable universal constant , and
. The previously known bounds for and are not
elementary recursive
On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations
This paper is mainly devoted to the study of the differentiation index and
the order for quasi-regular implicit ordinary differential algebraic equation
(DAE) systems. We give an algebraic definition of the differentiation index and
prove a Jacobi-type upper bound for the sum of the order and the
differentiation index. Our techniques also enable us to obtain an alternative
proof of a combinatorial bound proposed by Jacobi for the order.
As a consequence of our approach we deduce an upper bound for the
Hilbert-Kolchin regularity and an effective ideal membership test for
quasi-regular implicit systems. Finally, we prove a theorem of existence and
uniqueness of solutions for implicit differential systems
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Effective differential Lüroth's theorem
This paper focuses on effectivity aspects of the Lüroth's theorem in differential fields. Let F be an ordinary differential field of characteristic 0 and F〈u〉 be the field of differential rational functions generated by a single indeterminate u. Let be given non-constant rational functions v1,vn∈F〈u〉 generating a differential subfield G⊆F〈u〉. The differential Lüroth's theorem proved by Ritt in 1932 states that there exists v∈G such that G=F〈v〉. Here we prove that the total order and degree of a generator v are bounded by minjord(vj) and (n d(e +1) +1)2e +1, respectively, where e:=maxjord(vj) and d:=maxjdeg(vj). As a byproduct, our techniques enable us to compute a Lüroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.Fil: D'Alfonso. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Jeronimo, Gabriela Tali. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló". IMAS - CONICET; ArgentinaFil: Solernó, Pablo Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló". IMAS - CONICET; Argentina; Argentin
Quantitative aspects of the generalized differential Lüroth's Theorem
Let F be a differential field of characteristic 0, t=t1,…,tm a finite set of differential indeterminates over F and G⊂F〈t〉 a differential field extension of F, generated by nonconstant rational functions α1,…,αn of total degree and order bounded by d and e≥1 respectively. The generalized differential Lüroth's Theorem states that if the differential transcendence degree of G over F is 1, there exists v∈G such that G=F〈v〉. We prove a new explicit upper bound for the degree of v in terms of n,m,d and e. Further, we exhibit an effective procedure to compute v.Fil: D'Alfonso, Lisi. Universidad de Buenos Aires. Ciclo Básico Común; ArgentinaFil: Jeronimo, Gabriela Tali. Universidad de Buenos Aires. Ciclo Básico Común; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Solernó, Pablo Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
A decision method for the integrability of differential-algebraic Pfaffian systems
We prove an effective integrability criterion for differential-algebraic Pfaffian systems leading to a decision method of consistency with a triple exponential complexity bound. As a byproduct, we obtain an upper bound for the order of differentiations in the differential Nullstellensatz for these systems.Fil: D'Alfonso, Lisi. Universidad de Buenos Aires. Ciclo Básico Común; ArgentinaFil: Jeronimo, Gabriela Tali. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Solernó, Pablo Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin