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October 29[edit]

Positive-dimensional system of polynomial equations[edit]

System of polynomial equations#What is solving? 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)[reply]

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)[reply]

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#Positive-dimensional system of polynomial equations, where I also posted this question. Loraof (talk) 17:02, 3 November 2017 (UTC)[reply]

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

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)[reply]

Partial derivatives in medical education curricula[edit]

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)[reply]