Re: [sage-devel] Real algebraic varieties
On Sat, Dec 15, 2018 at 6:25 PM Thierry wrote: > > On Sat, Dec 15, 2018 at 05:58:57PM +, Dima Pasechnik wrote: > > On Sat, Dec 15, 2018 at 5:37 PM Thierry > > wrote: > > > > > > Hi, > > > > > > this question is related to the thread about Groebner bases. > > > > > > Are there some free-software implementations for real algebraic geometry > > > available somewhere ? Could Giac or Singular help with that ? Or maybe > > > Reduce or Macaulay2 (that are not shipped with Sage) ? > > > > > > More precisely, suppose i have a polynomial system of equations over QQ, > > > whose dimension (as the complex variety of an ideal) is positive. But i > > > know that the number of *real* solutions is finite. How to list them in > > > Sage ? I can easily get its Groebner basis, but not its real variety. > > > > If I am not mistaken (then what I propose is more of an heuristic), in > > such a case the real points are all on the singular locus of your > > variety. > > Thanks for you answer. > > Does Sage computes that locus easily ? not sure; theory for computing the singular locus is described in Sect 5.7 of "Singular introduction to commutative algebra", with Singular code how to do it - one need equidimensional components, and then minors of certain sizes of the Jacobian matrix will give you their singular loci. > > Ciao, > Thierry > > > > Compute it, hopefully it is 0-dimensional (otherwise, repeat), and > > select real points among all the points. > > > > > > > > > > RAGlib [1] seems to do that using Groebner bases, but it is a Maple^TM > > > package. > > > > > > The only thing i found within Sage is qepcad, but it is not powerful > > > enough to return anything. > > > > > > Ciao, > > > Thierry > > > > > > [1] https://www-polsys.lip6.fr/~safey/RAGLib/ > > > > > > -- > > > You received this message because you are subscribed to the Google Groups > > > "sage-devel" group. > > > To unsubscribe from this group and stop receiving emails from it, send an > > > email to sage-devel+unsubscr...@googlegroups.com. > > > To post to this group, send email to sage-devel@googlegroups.com. > > > Visit this group at https://groups.google.com/group/sage-devel. > > > For more options, visit https://groups.google.com/d/optout. > > > > -- > > You received this message because you are subscribed to the Google Groups > > "sage-devel" group. > > To unsubscribe from this group and stop receiving emails from it, send an > > email to sage-devel+unsubscr...@googlegroups.com. > > To post to this group, send email to sage-devel@googlegroups.com. > > Visit this group at https://groups.google.com/group/sage-devel. > > For more options, visit https://groups.google.com/d/optout. > > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To post to this group, send email to sage-devel@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-devel. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Real algebraic varieties
On Sat, Dec 15, 2018 at 05:58:57PM +, Dima Pasechnik wrote: > On Sat, Dec 15, 2018 at 5:37 PM Thierry > wrote: > > > > Hi, > > > > this question is related to the thread about Groebner bases. > > > > Are there some free-software implementations for real algebraic geometry > > available somewhere ? Could Giac or Singular help with that ? Or maybe > > Reduce or Macaulay2 (that are not shipped with Sage) ? > > > > More precisely, suppose i have a polynomial system of equations over QQ, > > whose dimension (as the complex variety of an ideal) is positive. But i > > know that the number of *real* solutions is finite. How to list them in > > Sage ? I can easily get its Groebner basis, but not its real variety. > > If I am not mistaken (then what I propose is more of an heuristic), in > such a case the real points are all on the singular locus of your > variety. Thanks for you answer. Does Sage computes that locus easily ? Ciao, Thierry > Compute it, hopefully it is 0-dimensional (otherwise, repeat), and > select real points among all the points. > > > > > > RAGlib [1] seems to do that using Groebner bases, but it is a Maple^TM > > package. > > > > The only thing i found within Sage is qepcad, but it is not powerful > > enough to return anything. > > > > Ciao, > > Thierry > > > > [1] https://www-polsys.lip6.fr/~safey/RAGLib/ > > > > -- > > You received this message because you are subscribed to the Google Groups > > "sage-devel" group. > > To unsubscribe from this group and stop receiving emails from it, send an > > email to sage-devel+unsubscr...@googlegroups.com. > > To post to this group, send email to sage-devel@googlegroups.com. > > Visit this group at https://groups.google.com/group/sage-devel. > > For more options, visit https://groups.google.com/d/optout. > > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To post to this group, send email to sage-devel@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-devel. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Real algebraic varieties
On Sat, Dec 15, 2018 at 5:37 PM Thierry wrote: > > Hi, > > this question is related to the thread about Groebner bases. > > Are there some free-software implementations for real algebraic geometry > available somewhere ? Could Giac or Singular help with that ? Or maybe > Reduce or Macaulay2 (that are not shipped with Sage) ? > > More precisely, suppose i have a polynomial system of equations over QQ, > whose dimension (as the complex variety of an ideal) is positive. But i > know that the number of *real* solutions is finite. How to list them in > Sage ? I can easily get its Groebner basis, but not its real variety. If I am not mistaken (then what I propose is more of an heuristic), in such a case the real points are all on the singular locus of your variety. Compute it, hopefully it is 0-dimensional (otherwise, repeat), and select real points among all the points. > > RAGlib [1] seems to do that using Groebner bases, but it is a Maple^TM > package. > > The only thing i found within Sage is qepcad, but it is not powerful > enough to return anything. > > Ciao, > Thierry > > [1] https://www-polsys.lip6.fr/~safey/RAGLib/ > > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To post to this group, send email to sage-devel@googlegroups.com. > Visit this group at https://groups.google.com/group/sage-devel. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
