# polynomial equation with non-integer powers

If for every $$t$$ $$\sum_{i=0}^{k_1}\left[\left(a_i^Tx-b_i\right)t^i\right]=0$$ where $$a_i \in \mathbb{R}^{n \times 1}$$, $$x \in \mathbb{R}^{n \times 1}, \forall i \in \{0,\dots, k_1\}$$, and $$b_i \in \mathbb{R}$$, then $$Ax=b$$ where the rows of $$A$$ are the vectors $$a_i^T$$ and the components of $$b$$ are the values $$b_i$$, $$\forall i \in \{0,\dots, k_1\}$$.

For the case $$\sum_{i=0}^{k_1}\left[\left(a_i^Tx-b_i\right)t^i\right]+t^\lambda\sum_{i=0}^{k_2}\left[\left(c_i^Tx-d_i\right)t^i\right]=0$$ where $$\lambda$$ is not an integer, can we form a similar system of equations? For example, $$\left[\begin{matrix} A \\ C \end{matrix}\right]x=\left[\begin{matrix} b \\ d \end{matrix}\right]$$ where the rows of $$C$$ are the vectors $$c_i^T$$ and the components of $$d$$ are the values $$d_i$$, $$\forall i \in \{0,\dots, k_2\}$$?

• so a Piseux series with infinitely many coefficients 0 ? – Roddy MacPhee Apr 18 at 19:26
• That's correct. – ApPs Apr 19 at 20:24