In a scientific article on the net, i find without further details, the following paragraph :
Let $ X \subset \mathbb{C} $ be un open connected subset, and $ x \in X $.
A linear differential equation of order $ n $ : $$ y^{(n)} + a_1 y^{(n-1)} + \dots + a_{n-1} y' + a_n = 0 $$ define a monodromy representation : $ \rho \ : \ \pi_1 (X,x) \to \mathrm{GL}_{n} ( \mathbb{C} ) $.
By the Cauchy existence theorem, the local solutions in a neighbourhood of $ x $ form a $ \mathbb{C} $ - vector space of dimension n endowed with monodromy action of $ \pi_1 (X,x) $.
- My questions are to know :
How is the monodromy representation $ \rho \ : \ \pi_1 (X,x) \to \mathrm{GL}_{n} ( \mathbb{C} ) $ defined, and, what is the intuition and the use of $ \rho $ compared to the linear differential equation : $ y^{(n)} + a_1 y^{(n-1)} + \dots + a_{n-1} y' + a_n = 0 $ ?
N.B. : The paragraph above is from the following pdf : http://www.galois.ihp.fr/wp-content/uploads/2011/12/T.-Szamuely.pdf , page : $ 48 $.
Thanks in advance for your help.