Jelly, (削除) 22 (削除ここまで) 17 bytes

×ばつ€Œd§ṙLC${Ṛð/ṫ‘}Ḣ

A dyadic link accepting a list of coefficient lists on the left and the exponent on the right.

Try it online! Or see the test suite.

Jelly, (削除) 22 17 (削除ここまで) 7 bytes

-10 bytes by realising I'd implemented convolution - go upvote Luis Mendo's MATL answer too, as that was what made me realise this.

æc/ṫ‘}Ḣ

A dyadic link accepting a list of coefficient lists on the left and the exponent on the right.

Try it online! Or see the test suite.

How?

æc/ṫ‘}Ḣ - Link: list of lists, Ps; integer, E
 / - reduce (Ps) by:
æc - convolution
 } - use right argument (E) for:
 ‘ - increment -> E+1
 ṫ - tail from (1-indexed) index (E+1)
 Ḣ - head (if given an empty list yields 0)

The 17 (without the convolution atom) was:

×ばつ€Œd§ṙLC${Ṛð/ṫ‘}Ḣ

Jelly, 22 bytes

×ばつ€Œd§ṙLC${Ṛð/;0ẋ$}ị@‘}

A dyadic link accepting a list of coefficient lists on the left and the exponent on the right.

Try it online! Or see the test suite .

Feels too long; 5 bytes used just to handle the higher than maximal coefficient cases.

Jelly, (削除) 22 (削除ここまで) 17 bytes

×ばつ€Œd§ṙLC${Ṛð/ṫ‘}Ḣ

A dyadic link accepting a list of coefficient lists on the left and the exponent on the right.

Try it online! Or see the test suite.

Jelly, 22 bytes

×ばつ€Œd§ṙLC${Ṛð/;0ẋ$}ị@‘}

A dyadic link accepting a list of coefficient lists on the left and the exponent on the right.

Try it online!

Feels too long; 5 bytes used just to handle the higher than maximal coefficient cases.

Jelly, 22 bytes

×ばつ€Œd§ṙLC${Ṛð/;0ẋ$}ị@‘}

A dyadic link accepting a list of coefficient lists on the left and the exponent on the right.

Try it online! Or see the test suite .

Feels too long; 5 bytes used just to handle the higher than maximal coefficient cases.