Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through $$ p(y) = t_1 y + t_2 y^2 + \cdots + t_k y^k \in A\,. $$

Let $I$ be the ideal in $A$ generated by $\{p(y)^2 \mid y \in \mathbb{R}\}$, and let $S = A/I$ be the quotient algebra of $A$ by $I$.

Has $S$ been studied? Does it have a name?