Thanks Nils for these interesting solutions. To answer your question, I am only interested in real solutions. Yes, Wolfram Alpha may only show some solutions. I used it to determine if there were ANY solutions, which there were, so that I might expect some solutions rather than no solutions from Sage. The equations provided were test equations, in preparation for using Sage to solve other systems of equations for which I seek solutions. My purpose in presenting my question to the group is to ascertain whether the method by which I have presented the equations to Sage for solution is correct, and why no solutions were forthcoming. Thanks again.
On Saturday, August 20, 2016 at 9:27:34 PM UTC-7, Nils Bruin wrote: > > On Saturday, August 20, 2016 at 8:29:02 PM UTC-7, Roger S wrote: >> >> Hello, >> >> I'm new to Sage and your help is appreciated. I am trying to find >> numerical solutions for a system of three equations in three unknowns as >> follows (the coefficients here are arbitrary for testing purposes, but I >> have verified that there are in fact numerical solutions): >> >> xyz + 2xy + 3yz + 4xz + 5x + 6y + 7z = 0 >> 8xyz + 9xy + 10yz + 11xz + 12x + 13y + 14z = 0 >> 15xyz + 16xy + 17yz + 18xz + 19x + 20y + 21z = 0 >> >> In response to "solve" Sage is returning the original equations with no >> numerical results. The actual solutions (from Wolfram Alpha) are: >> >> x=-3, y=-3/4, z=-3 and x=-4/3, y=8, z=-4/3 >> > > Why do you think that's the "correct" answer? Why is x=y=z=0 an invalid > solution? > > In actuality, your equations happen to describe a positive-dimensional > solution set; it happens to be an algebraic curve of genus 1 > (congratulations; not all intersections of 3 cubic equations in 3-space > yield genus 1 curves!), and this one happens to have infinitely many > rational solutions. Wolfram Alpha missed this one too, for instance: > > > (197645777049634548512744383174389298033637463502382310142600087646802794602853\ > > 1531257080273157657786059636413352036993143703588065912642/86947948899367165333\ > > 5402658211209586423986666805945772208167570149400681547564137405585160928838798\ > 687063936496966509397849639384765441, > -1287675836873589883723142251933763847414\ > > 8438738177920292488334858041920061261361870906384138469498407486400459861979322\ > > 078869377962299641984/100643168107472738754649925590736684519831838241166038091\ > > 4129700864889555666628790657398853137957618857694788459219913260417008953931278\ > 65833, > -97036485140207458622029618188063119110557273624372030829814799917732925\ > > 258180390881114790660411640849323543230464453805657775708915879841278/149365077\ > > 9384072663572446521641658389504377145205608740382948594185066596356570619651876\ > 87554301362956629570143317207424117326061853527647709) > > You also haven't stated what kind of solutions you are interested in. Only > rational ones? or is any real or complex-valued solution OK? > > It's not always so straightforward to describe a solution set in a way > different from a tautological description. This is a mathematical problem, > not just one for computer algebra systems. > > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at https://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
