Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that there is a map of $R$-modules $\varphi: S \to R$ that preserves degrees and takes each element of ...

This is testing my understanding of algebras, subalgebras, & polynomial rings. Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that there is a map of ...

Here is the notation & motivation.: Conjecture: If $f \in \hat{m}$, then $f = \sum g_i f_i$ where each $g_i$ is homogeneous. I've tried proof by induction on degree of $f$, number of generators making up $f$, breaking up the $f_i$ into degree buckets. What am I not seeing?

I think this is a very basic question but somehow I am unable to understand the answer. Find the number of zeroes of $f(x) = x^3 + x + 1$. Answer in the book : $f^\prime (x) = 3x^2 + 1$ since $3x^2 + 1 \ge 0$ for all $x \in \mathbb{R}$ therefore $f^\prime (x) \ge 1 $ for all $x \in \m...

Find number of zeroes of $f(x) = 1 - x^{-2}$. I assume that this function has two or more zeroes in the domain $ \mathbb{R} - \{0\}$. Since $f^\prime (x) = \large{2\over x^3}$, therefore we can say $f^\prime(x) \ne 0$ for all $x \in \mathbb{R} - \{0\}$. Therefore by Rolle's theor...

Suppose that, $$f(x)=\sum_{n=0}^{\infty}a_nx^n$$ converges on $(-R,R)$ for some $R>0$. If we let $(x_n)$ be a sequence in $(-R,R)$ with $x_n\ne 0$ but $limx_n=0$. If $f(x_n)=0$ for all $n\in N$, I need to show that $f(x)=0$ for all $x\in (-R,R)$. My thinking is that I could use Rolle's Theorem...

There are separate tag connectedness and path-connected. In the tag-info cor connectedness it is explicitly mentioned that it includes (among other thing) questions about path-connectedness. (Moreover, having too many closely related tags causes problem, since there is a limit at most 5 tags per ...