6.3 The folded case ( s > 1)
6.3.3 The general case
When the regularity assumption of the previous subsection does not hold, every
step will entail a division by zero, so that the algorithm does not do any sensible computation and just return a series with indeterminate coefficients, and that the post-processing phase will have to do all the work. In this subsection, we give an algorithm inspired by the Newton-Puiseux algorithm presented before, that can han- dle such situations better (although some steps will still require the introduction of indeterminates).
In [2], Beelen and Brander gave a recursive algorithm of Hensel lifting that can compute f, inspired by Roth and Ruckenstein’s algorithm; at steps where the value of the coefficient fi cannot be determined, Beelen and Brander test all elements of
the underlying field (which is similar to our use of indeterminate coefficients for such steps). Compared to that algorithm, the main difference in ours lies in the use of a Newton polygon construction, making our algorithm an extension of the Newton- Puiseux algorithm presented before for the case s= 1.
Let Q be a polynomial inK[x, z1, z2, . . . , zs], which we write as
Q= X (α,ρ)∈SQ
aα,ρxαz1ρ1z
ρ2
2 . . . zρss,
where SQ is the support of Q, each aα,ρ is in K− {0} and ρ = (ρ1, . . . , ρs). Let us
define the points of Q to be the set
where we write
Pα,ρ= (α, ρ1+ρ2+· · ·+ρs)∈N2.
The Newton polygon N(Q) of Q is the convex hull of the set [
P∈P(Q)
(P +N2) Nis integer ring;
the slopes of this polygon are defined as in the previous section. Figure 6.4 gives an example of a Newton polygon of a equation having three variables.
x4−x3y2−x3y−x3z2−x3z+x2y3+x2y2z2+x2yz+ 1 2 3 4 5 6 7 1 2 3 4 5 6 x y x2yz2+x2yz+x2z3−xy3z2−xy3z−xy2z3−xyz3+y3z3 Newton polygon for the equation :
Figure 6.4: Newton polygon
Inspired by the Newton-Puiseux algorithm given before, Algorithm15below com- putes the power series roots of Q using its Newton polygon. As before, the compu- tation is based on the geometry of the Newton polygon; this time however, not only the edges but also the vertices of the Newton polygon may contribute to the solutions (this is because in this 2-dimensional representation, several monomials of Q may correspond to a given vertex). Thus, for λ ∈N, we define the following:
• L(Q, λ) is the straight line with slope −1
λ which intersects N(Q) at either a
vertex or an edge;
N(Q); this is a polynomial in w, which (when nonzero) can be seen as the coefficient of the lowest degree term in x of
Q(x, xλw,(γx)λw, . . . ,(γs−1x)λw),
or equivalently of
Q(x, f(x), . . . , f(γs−1x)) for f of the form f =xλ(w+w′x+w′′x2+· · ·).
Remark that the nonzero roots of ΦQ,λ give the possible values w such that our
equation may admit a solution of the formf =xλ(w+w′x+w′′x2+· · ·). As a small example, consider the polynomial Q =z1 −z2, whose Newton polygon has only one vertex (0,1) and γ =−1. Whenλ = 1, we obtain ΦQ,1 = 2w, which has no nonzero root; this indicates that there is no power series solution f of f(x) = f(−x) of the form wx+· · ·, for w nonzero. On the contrary, for λ = 2, we find ΦQ,2 = 0; this shows that, as far as the lowest coefficient in concerned, any w may be suitable for a solution f of f(x) =f(−x) of the formwx2 +· · ·
As this example shows, in this algorithm, as in the cases of the two previous subsections, we may of course come up with outputs f that have indeterminates as coefficients. When this is the case, we use the same post-processing step as in the previous subection.
Correctness follows from the discussion preceding the algorithm; alternatively, one may consult [9], where a more complex algorithm is given (to compute more general solutions than power series).
To conclude, let us mention an analogue to the last remark of the previous section. As we noted there, after sufficiently many initial steps, the solutions get “separated”, and the coefficients fi become uniquely determined. A similar phenomenon happens
in the folded case, up to a minor modification; to describe it, we follow Cano and Fortuny Ayuso [9].
Given a solution f(x) = Pi>0fixµi ∈ K[x] of (6.1), define the sequence of poly-
nomialsQ0 =Q and, for i>0,
Qi+1 =Qi(x, fixµi +z1, . . . , fiγ(s−1)ixµi +zs).
These polynomials are essentially the ones seen during the Newton-Puiseux algorithm (up to change of variables of the formx7→xηx, for suitable values ofη), but are better
Algorithm 15: Newton-Puiseux expansion(Q, i, k, ψ)
1: if Qis of the form czρ1
1 · · ·zsρs for some cinK then
2: outputψ 3: end if
4: if i>k then 5: outputψ 6: end if
7: replace Q by Q/xr, where r is the largest integer such that xr divides Q 8: for λ=i, . . . , k−1 do
9: if ΦQ,λ(w) = 0then
10: ψ[i+λ] = wi (wi is a placeholder for the coefficient) 11: Qe=Q(x, xλ(wi+z1), γxλ(wi+z2), . . . , γs−1xλ(wi+zs)) 12: Newton-Puiseux expansion(Q, ie +λ, k, ψ)
13: else
14: let ℘ be the roots in K of ΦQ,λ 15: for each root ς 6= 0∈℘ do 16: ψ[i+λ] =ς 17: Qe =Q(x, xλ(ς +z1), γλxλ(ς +z2), . . . , γ(s−1)λxλ(ς+zs)) 18: Newton-Puiseux expansion(Q, ie +λ, k, ψ) 19: end for 20: end if 21: end for
For anyi>0, letpi= (αi, βi) be the point with highest ordinate at the intersection
of the line L(Qi, i) and the Newton polygon of Qi. Cano and Fortuny Ayuso proved
the following:
• the sequence βi is non-decreasing;
• there exists i0 >0 such that for i>i0, βi =βi0;
• xαizρ1
1 · · ·zsρs appears with a nonzero coefficient in Qi, with ρ1 + · · · +
ρs = βi and ρj > 1 for some index j, then f is also a root of R =
∂ρ1+···+ρs−1Q/∂zρ1
1 · · ·∂z
ρj−1
j · · ·∂zρss;
• if in addition αi = 0, then R satisfies the assumptions of Subsection 6.3.2
In other words, up to replacing Q by a well-chosen derivative, we are reduced to the regular case; recall however that even in the regular case, some coefficients fi of f
may still be undertermined for those indices ithat cancel the critical equation. The point pi0 is called the pivot pointassociated to Q and f.