Search Results

Here is a comment relating the question to representations of sums of five squares of rational polynomials (sorry, too long for the comment field): Suppose that $f$ is a linear combination of two squ …

Over the complex numbers, connected linear algebraic groups correspond to Lie algebras in the usual way. This Lie correspondence breaks down over number fields, and breaks down even more over fields o …

Even for commutative associative algebras it is not true. The article of Björn Poonen "Isomorphism types of commutative algebras of finite rank over an algebraically closed field" gives a classificati …

The algebra $S$ is etale, i.e., smooth of relative dimension $0$. A proof, that $S$ is flat over $R$ can be found in www.math.purdue.edu/~dvb/preprints/etale.pdf‎, see Proposition $1.3.5$ on page $1 …

Yes, the orthogonal group makes sense over any field $k$. It is an linear algebraic group. In fact the theory of linear algebraic groups generalizes that of linear Lie groups over the real or complex …

In general, there is the following result on upper bounds for the degree of elements in the (reduced) Groebner basis: Let $G$ be a reduced Groebner basis of an ideal $I=\langle f_1,\ldots, f_r\rangle …

One "arithmetic version" of the Nullstellensatz states that if $f_1, ..., f_s$ belong to $\mathbb{Z}[X_1,...,X_n]$ without a common zero in $\mathbb{C}^n$, then there exist $a \in \mathbb{Z} \setminus …

Grothendieck expected the moduli spaces $\mathcal{M}_{g,n}$ over $\mathbb{Q}$ to be the basic examples of anabelian varieties (besides hyperbolic curves, which was proved by Mochizuki, even over numbe …

I found a proof of the following fact in the article of G. Payne and L. Vaserstein, "Sums of three cubes", contained in the book "The arithmetic of function fields" (1992): The set of integral solutio …

It is, for example, explained in paragraph $8$ (Appendix) of the book Unipotent Algebraic Groups, Lecture Notes in Mathematics Volume 414, 1974, by Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhi …

Over the complex numbers, there are several normal forms. A ternary cubic can be written as $$ \lambda XYZ-(X+Y+Z)^3 $$ with a parameter $\lambda$. This normal form is invariant under the symmetric gr …

Lie’s theorem indeed still holds in positive characteristic provided the dimension of the vector space is less than the characteristic. For reference see, for example, the remark before example $81$ i …

In the lecture notes Albert algebras by H.P. Petersson, written in $2012$ this is still mentioned as an open problem, see Question $13.2$: Is an Albert algebra $J$ determined up to isomorphism by its …

Using Groebner bases (or even direct computing) we see, that the solutions of your system $f_5=f_6=0$ are as follows: Case 1: $a_1b_2-a_2b_1=0$. Then it follows $y^2+z^2=0$. Over the real numbers th …

I am by no means an expert here, so this is just a "long comment" regarding the question why analytic tools come in. In algebraic geometry intersection theory over complex numbers is a powerful tool t …