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40 CHAPTER 2. POLYNOMIALS
Figure 2.4: DAG representation
1 x
↘ ↙
p → + ← q
⇓
r → ∗
Figure 2.5: Tree representation
1 x 1
↘ ↙ ↘ ↙
p → + + ← q
↘ ↙
r → ∗
reader to [BFSS06]. We will also only consider monic polynomials.
Notation 13 Let p = x n + ∑ n−1
i=0 a)ixi = ∏ n
i=0 (x−α i) be a polynomial of degree
n. Let β s = ∑ n
∑
i=0 αs i , and define the Newton series of p to be Newton(p) =
s≥0 β sT s .
It is well-known that the a i and α i are related:
a n−1 = −
a n−2 =
These are then related to the β i :
n∑
n∑
i=0
α i
n∑
i=0 j=i+1
α i α j
. . .
a 0 =
∏
n
(−1) n α i .
i=0
β 1 = −a n−1
β 2 1 = β 2 + 2a n−2 .
.
..
Hence, in characteristic 0, the β i (i ≤ n) form an alternative to the a i , a
fact known since 1840 [LV40]. But how do we convert rapidly between these
representations?