I would try keeping the grass well fertilized AND well watered. Those roots are taking up a lot of the nutrients in that soil. And also taking up a lot of the water. Not only that but the leaf canopy is keeping rain water from penetrating under the "umbrella" they create.
I would reseed a good grass seed for your area. And I would cover that seed with about an inch of mushroom compost. That compost will keep the seed moist while the seeds germinate, and then will compliment the fertilizer you add. Don't let the seed bed dry out but don't make it swampy.
I have a cluster of a half dozen Norway Maples. The grass grows and I only fertilize with the clippings that I don't remove. We use the area near there for a turn around and the only problem areas are where the traffic is. I also have Norway Maples along my driveway and the grass grows fine. If you have ruts between the visible roots you should fill those ruts before seeding. I'm talking where the ground is lower than neighboring areas, not the areas lower than the roots.
If you're using a grass mixture with yellow fescue that grass grows well in shade, the reason it's in the mix. However it doesn't do well in dry conditions, like under trees.
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