By Boyer Ch. P.

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**Additional info for 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients**

**Sample text**

E. that all the coeﬃcients of A and B are positive. Let C = A B = c p+ q X p+ q + + c0. It is clear that all the coeﬃcients of C are positive. It remains to prove that c2k ck−1 ck+1. 2 Real Root Counting Using the partition of (h, j) ∈ Z2 h > j and {(h, h − 1) h ∈ Z}. 49 in (j + 1, h − 1) ∈ Z2 h F F j F c2k − ck−1 ck+1 = ah a j bk−h bk− j + h j ah a j bk−h bk− j h>j − ah a j bk−h+1 bk− j −1 − h j = ah a j bk−h bk− j + h j a j +1 ah−1 bk− j −1 bk −h−1 h + ah a j bk −h+1 bk −j −1 h> j j ah ah−1 bk−h bk−h+1 − ah ah−1 bk−h+1bk−h h h − ah a j bk−h+1 bk− j −1 − h = j a j +1 ah−1bk −j bk−h h j (ah a j − ah−1 a j +1) (bk −j bk−h − bk− j −1 bk −h+1).

2 Real Root Counting Using the partition of (h, j) ∈ Z2 h > j and {(h, h − 1) h ∈ Z}. 49 in (j + 1, h − 1) ∈ Z2 h F F j F c2k − ck−1 ck+1 = ah a j bk−h bk− j + h j ah a j bk−h bk− j h>j − ah a j bk−h+1 bk− j −1 − h j = ah a j bk−h bk− j + h j a j +1 ah−1 bk− j −1 bk −h−1 h + ah a j bk −h+1 bk −j −1 h> j j ah ah−1 bk−h bk−h+1 − ah ah−1 bk−h+1bk−h h h − ah a j bk−h+1 bk− j −1 − h = j a j +1 ah−1bk −j bk−h h j (ah a j − ah−1 a j +1) (bk −j bk−h − bk− j −1 bk −h+1). h j Since A is normal and a0, , a p are positive, one has a p−1 ap a p−2 a p−1 a0 , a1 and ah a j − ah−1 a j+1 0, for all k j.

This can be done through the notion of virtual roots. 50 2 Real Closed Fields The virtual roots of P will enjoy the following properties: a) the number of virtual roots of P counted with virtual multiplicities is equal to the degree p of P , b) on an open interval deﬁned by virtual roots, the sign of P is ﬁxed, c) virtual roots of P and virtual roots of P are interlaced: if x1 ≤ ≤ x p are the virtual roots of P and y1 ≤ ≤ y p−1 are the virtual roots of P , then x1 ≤ y1 ≤ ≤ xp−1 ≤ y p−1 ≤ x p. Given these properties, in the particular case where P is a polynomial of degree p with all its roots real and simple, virtual roots and real roots clearly coincide.

### 3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients by Boyer Ch. P.

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