I am self studying a research paper in analytic number theory.
A part of the paper uses Linear Algebra. Actually, I am not good in Linear Algebra ( during the teaching of Linear Algebra course I was self studying a course not part of my syllabus) .
So, can anybody please tell me how to deduce these statements.
Statement let $D$ be the product of all primes $\geq (1-2\epsilon) \log (s) $. Let $\delta$ be the number of divisors of $D$ . Let $i_0=1$ and all $ i_j$ are odd and $3= i_1 < i_2<\cdots < i_{\delta-1}\leq s$ be odd integers
Consider set $\mathcal D$ of all divisors of $D$ so that $\lvert\mathcal D\rvert = \delta$ . Assume that matrix $[ d^{i_j}]_{d\in \mathcal D , 0\leq j \leq \delta -1 }$ is invertible. ( Note the given matrix is Vandermonde Matrix)
Then authors (using linear algebra I suppose) write that there exist $w_d\in\Bbb Z$, where $d\in\mathcal D$, such that $\sum_{d\in\mathcal D} w_d d^{i_j} =0$ for all $j\in \{1, 2,...,\delta-1\}$ (condition 1) and $\sum_{d\in\mathcal D}w_d d \neq 0 $ (condition 2)
I am unable to get to these deductions .
Please help.
You may consider the system of $\delta$-many $\Bbb Q$-linear equations in $\delta$ unknowns $(W_d\,:\, d\in\mathcal D)$, where the $j$-th equation is $\sum_{d\in \mathcal D}W_dd^{i_j}=0$ for $j=1,\cdots,\delta-1$ and $\sum_{d\in\mathcal D}W_dd=1$ for $j=0$. The coefficient matrix of this system is $A_{j,d}:=d^{i_j}$, with $d$ ranging over $\mathcal D$ and $j$ ranging in $\{0,\cdots,\delta-1\}$. Since we've already established that the coefficient matrix is invertible, this system has exactly one solution $(W_d\,:\, d\in\mathcal D)\in \Bbb Q^{\mathcal D}$. Now, multiply by a common denominator all the $W_d$-s and you'll obtain integer values of $w_d$ such as the ones you are looking for: condition (2) remains the same, and condition (1) comes from multiplying both sides by a non-zero integer.