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0081306187557833118)t5 − 1 accounting for 2 × 5 = 10 remaining solutions. The total number of solutions is 70, as indicated by the mixed volume computation. e. 0081306187557833118. 2 A General Approach That the ﬁrst n − 1 equations of cyclic n-roots system give explicit solution lines is exceptional. For general polynomial systems we can use the leading term of the Puiseux series to compute witness sets [49] for the space curves deﬁned by the ﬁrst n − 1 equations. Then via the diagonal homotopy [48] we can intersect the space curves with the rest of the system.

I Prilozhen 9(3), 1–4 (1975) 9. : Methods to “divide out” certain solutions from systems of algebraic equations, applied to ﬁnd all cyclic 8-roots. In: Herman, I. ) The Use of Projective Geometry in Computer Graphics. LNCS, vol. 564, pp. 57–70. Springer, Heidelberg (1992) 10. : All cyclic p-roots of index 3, found by symmetrypreserving calculations. dk/~ haagerup 11. : Computing tropical varieties. Journal of Symbolic Computation 42(1), 54–73 (2007) 12. : Power Geometry in Algebraic and Diﬀerential Equations.

Let hR (X1 ) be the product of the initials of the polynomials in R. Let ρ > 0 be small enough such that the set Uρ := {x = (x1 , . . , xs ) ∈ Cs | 0 < |x1 | < ρ} does not contain any zeros of hR . Deﬁne Vρ (R) := V (R) ∩ Uρ . Then, we have W (R) ∩ Uρ = Vρ (R). Proof. It follows from the observation that Uρ ∩ V (hR ) = ∅. Notation 1. Let W ⊆ Cs . Denote lim0 (W ) := {x = (x1 , . . , xs ) ∈ Cs | x ∈ lim(W ) and x1 = 0}. Lemma 2. We have lim0 (W (R)) = lim0 (Vρ (R)). Proof. By Lemma 1, we have W (R) ∩ Uρ = Vρ (R).