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Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{... 2answers 632 views ### Differential ideal membership problem We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case? To be ... 1answer 1k views ### Existence of a real-valued solution to system of multivariate polynomial equations Given a system of multivariate, polynomial equations, is there a way to determine if it has a solution in a given field (for instance the set of all reals). I don't care what the solution is, I just ... 2answers 565 views ### Dimension of a homogeneous polynomial system Let m\geq4 be an even integer, V\subset\mathbb{C}^{m-1} be the solution set of the following polynomial equations: \begin{cases} &\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}... 0answers 77 views ### Polynomial constraints triggered by irreducibility [closed] I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic:$$af^2 + bf + c = 0$$If we're working in a ring, ... 0answers 90 views ### Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials Let R=\mathbb{R}[X_1,\dots,X_n], and$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\} ...
I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques. For example, consider the equation $qn = mf$, where each of the variables ...