Wikipedia:Reference desk/Archives/Mathematics/2017 October 29

= October 29 =

Positive-dimensional system of polynomial equations
System of polynomial equations says

''If the system is positive-dimensional, it has infinitely many solutions. It is thus not possible to enumerate them. It follows that, in this case, solving may only mean "finding a description of the solutions from which the relevant properties of the solutions are easy to extract".''

If the multivariate polynomials in the equations share a common factor, then equating that factor to 0 gives a characterization of solutions of the system. But what if the polynomials in the system are all irreducible – can the system still be positive-dimensional? If so, what is an example? Loraof (talk) 18:53, 29 October 2017 (UTC)
 * As I read your question, a very simple example is the polynomial equations x=0 & y=0 considered as a system of polynomial equations in 3 variables (z too). These polynomials define the z-axis as the solution set, obviously one dimensional. In general, an Underdetermined system with more variables than equations is either inconsistent or has a solution set with dimension (number of variables) - (number of equations).John Z (talk) 01:40, 3 November 2017 (UTC)


 * Thanks. I should nave been more specific by specifying “non-constant polynomials”, which would preclude x=0, y=0; and I should have specified that I had in mind systems with the same number of equations as unknowns. For interested readers, a good example satisfying these conditions is given by D.Lazard in the last paragraph at Talk:System of polynomial equations, where I also posted this question. Loraof (talk) 17:02, 3 November 2017 (UTC)


 * Three planes intersecting in the same line? --JBL (talk) 12:30, 5 November 2017 (UTC)


 * Right, that’s a good linear example, although I was wondering about the case of greater than first degree. Your example brings out the broader point that if you start with two polynomials in three variables with a solution, and hence with an infinitude of solutions, you can take a weighted average of those polynomials to be the third polynomial, and no solutions are lost, so it’s 3×3 and positive dimensional. And likewise for higher dimensions. Loraof (talk) 15:17, 5 November 2017 (UTC)

Partial derivatives in medical education curricula
What is the extent of inclusion of advanced mathematical topics like partial derivatives and partial differential equations in usual medical education curricula? (Thanks.)--82.137.14.137 (talk) 23:45, 29 October 2017 (UTC)