Full discretization of distributed control problems for parabolic equations
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Full discretization of distributed control problems for parabolic equations Franck BOYER? joint work with Florence HUBERT? and Jerome LE ROUSSEAU† ? LATP, Aix-Marseille Universite †MAPMO, Universite d'Orleans IHP, November 2010 1/ 58 F. Boyer Control of full-discrete parabolic equations
Voir
- control problem
- equations
- aix-marseille universite
- full-discrete parabolic
- time discretization schemes
Motivation Factorization of cyclotomic polynomials Some steps to abelian extensions Conclusion
polynomials over finite nisticopylimenomial-t
Factorization of determi
Ivan Boyer
Doctorant sous la direction de Jean-François Mestre Institut Mathématique de Jussieu
AGCT-13 — C.I.R.M. March 15, 2011
I. Boyer
Factorization inFp[X]
fields
in
Motivation Factorization of cyclotomic polynomials Some steps to abelian extensions Conclusion Algorithmic aspect.
Context Schoof’s algorithm
Remark Thedeterministicaspect iscrucialin this talk : everything be-comes “trivial” in probabilistic time. In the same way, assuming G.R.H. would withdraw some of the interest of the following !
I. Boyer
Factorization in
Fp[
X]
Motivation Factorization of cyclotomic polynomialsContext Some steps to abelian extensionsSchoof’s algorithm Conclusion Factorization inFp[X]– Square roots in
Fp.
IThere are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. INo deterministic polynomial-time algorithm is known for factorization inFp[X]. Even in degree 2 ! IEasy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) IA lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. IHowever, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.
I. Boyer
Factorization inFp[X]
Motivation Factorization of cyclotomic polynomialsContext Some steps to abelian extensionsSchoof’s algorithm Conclusion Factorization inFp[X]– Square roots in
I
I
I
I
I
Fp.
There are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. No deterministic polynomial-time algorithm is known for factorization inFp[X] in degree 2 !. Even Easy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. However, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.
I. Boyer
Factorization inFp[X]
Motivation Factorization of cyclotomic polynomialsContext Some steps to abelian extensionsSchoof’s algorithm Conclusion Factorization inFp[X]– Square roots in
I
I
I
I
I
Fp
.
There are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. No deterministic polynomial-time algorithm is known for factorization inFp[X]. Even in degree 2 ! Easy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. However, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.
I. Boyer
Factorization inFp[X]
Motivation Factorization of cyclotomic polynomialsContext Some steps to abelian extensionsSchoof’s algorithm Conclusion Factorization inFp[X]– Square roots in
I
I
I
I
I
Fp
.
There are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. No deterministic polynomial-time algorithm is known for factorization inFp[X]. Even in degree 2 ! Easy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. However, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.
I. Boyer
Factorization inFp[X]
