Well, I dont know for a fact - but kadri gopalnath's sax playing received a lot of jazz musician's praise. Becoz we carnatic folks have nothing written, and we improvise upon one basic theme. So, in those lines, we can give a jazz feel to carnatic music as well. Thats what i feel, once again - please differ with me. After this I will try to provide same kinds of stuff in jazz like same song played by different people.
Chords just lays a foundation for a melody to improvise upon.
Play a major, then add seventh, then add ninth and then thirteenth and so on. Try singing a melody on top of that. You will find thats exactly what a jazz musician does. So my point is, there is definitely improvisations in jazz. Now speaking of carnatic music - you people mustve definitely heard fusions of jazz and carnatic.