Consider a polynomial system F =G= 0 in K[X, Y] and an isolated solution (x∗, y∗) of it. Most extensions of Newton iteration to the case where (x, y) has multiplicityM >1 seek to replace the given system with a new one, sayψ, such that the multiplicity ofψ at the root (x, y) is less than M – eventually, we reachM = 1, where we can apply Newton iteration without difficulty. Such a process is called deflation.
Following Lecerf’s approach, the deflated systems are constructed by considering suitable derivatives of the given system hF, Gi. For the complexity analysis, we will need to set a bound on the order of partial derivatives. The following construction as- signs to an isolated solution (x, y) of F = G = 0 a signature σ(x∗, y∗), of the form
σ(x∗, y∗) = (m,H, n, a,K). In essence, this signature predicts which derivatives of F, G
should be taken to reach a deflated ideal ψ satisfying the multiplicity reduction require- ment.
The following deflation lemma is the key to this construction. It follows very closely Lemma 4 from [40]. That reference deals with systems in an arbitrary number of variables, but relies on a generic change of variables, which we avoid here (by slightly changing the definition of integer m below).
Lemma 14(Deflation lemma). LethF, Gibe an ideal inK[X, Y], withF andGof degree at most d, and let (x∗, y∗)∈K2 be an isolated root ofhF, Gi with multiplicity M. Define
m:= min µ: ∂ µF ∂Yµ(x ∗ , y∗)6= 0 or ∂ µG ∂Yµ(x ∗ , y∗)6= 0 .
If in addition K has characteristic at least d, then: (a) m >0;
(b) m≤d;
(c) (x∗, y∗) is a root of ψ with multiplicity n, for some integer n satisfying 1 ≤ n ≤
M/m, where ψ := F, G,∂F ∂Y , ∂G ∂Y ,· · · , ∂m−1F ∂Ym−1, ∂m−1G ∂Ym−1 .
Proof. Upon translating the origin to (x∗, y∗), we can assume without loss of generality
that x∗ =y∗ = 0. To prove the first item, note thatF(0,0) = G(0,0) = 0, which implies
m >0. Let us further denote byI the ideal hF, Gi.
Let us next prove that m is finite. If all partial derivatives of F with respect to
Y, Y2, . . . , Yd vanish at (0,0), F(0, Y) must be the zero polynomial (recall that F has
degree at most d), so that X divides F. If this is the case for Gas well, X divides both
F and G, so (0,0) is not an isolated solution of F =G= 0, a contradiction. Using the following facts,
∂µF ∂Yµ(0,0) = 0 and ∂µG ∂Yµ(0,0) = 0, 0≤µ≤m−1 and ψ = F, G, ∂F ∂Y , ∂G ∂Y ,· · · , ∂m−1F ∂Ym−1, ∂m−1G ∂Ym−1 ,
it is clear that (0,0) is a root of ψ, so n ≥1. It remains to give an upper bound on it. We are going to work locally, by looking at F, G and their derivatives in K[[X, Y]]. So, by the definition of multiplicity, we have
M = dimKK[[X, Y]]/I and n= dimKK[[X, Y]]/ψ.
We are going to describe more precisely these residue class rings. Let us endowK[[X, Y]] with the order defined by
One verifies that this order is compatible with multiplication, and that 1> X and 1> Y
both hold. This is thus a local monomial order, as in [21, Chapter 4]. To any power series S in K[[X, Y]], we can associate its leading monomial lm(S) with respect to this order; this notation carries over to ideals in K[[X, Y]].
From [21, Theorem 4.3], we infer that the monomials in lm(I)cand lm(ψ)c– where the
exponentcdenotes complement – form bases of respectively
K[[X, Y]/I and K[[X, Y]/ψ.
In particular, the numbers of these monomials are respectively M and n. Define
T :={XaYm−1 ∈lm(I)c|a ≥0}.
Because lm(I) is stable by multiplication, for each element XaYm−1 of T, all monomials XaYb, for 0 ≤ b ≤ m−1, are in lm(I)c, whence M = |lm(I)c| ≥ m|T|. We now prove
that n is at most |T|.
• The definition of m implies that at least one of ∂∂YmmF or
∂mG
∂Ym does not vanish at
(0,0); let us assume without loss of generality that this is the case for ∂Y∂mmF. This
implies that for b = 0, . . . , m−1, the coefficient of the monomial Yb in F is zero,
while that ofYm is nonzero. The definition of our local order then implies thatYm
is the leading term of F. Thus, Ym is in lm(I), so that XaYm is in lm(I) for any
a≥0.
Consider an element P ∈ K[[X, Y]] having leading monomial XaYm. Because
m ≤d, and due to our assumption on the characteristic of K, we deduce that the leading monomial of ∂∂Ym−m1−P1 ism!X
aY. This shows that fora≥0,XaY is in lm(ψ).
• Similarly, let a0 be the smallest integer such that Xa0Ym1−1 is in lm(I); thus, a0 =|T|. Differentiatingm−1 times as above, we deduce that Xa0 is in lm(ψ).
The two items above prove that lm(ψ)c is contained in {Xa | 0 ≤ a < a
0}, so it has
cardinality at most a0 =|T|. This proves the lemma.
This lemma allows us to define the first components m,H of σ(x∗, y∗):
• m is defined as in the lemma;
• the string H∈ {”F”,”G”} indicates which of ∂mF
∂Ym and ∂ mG
∂Ym is nonzero at (x
∗, y∗); in
case of a tie, for definiteness, we choose F.
Using the same notation as above, let us define H = F (if H = ”F”) or H = G (if
H= ”G”), so that m= min µ: ∂ µH ∂Yµ(x ∗, y∗)6= 0
and (x∗, y∗) is a root of ∂m−1H ∂Ym−1.
Define Hc (the “complement” of H) as either Hc = G if H = F and Hc = F if
H =G. We could replace the system (F, G) by (∂∂Ym−m1−H1, H
c) to find the root (x∗, y∗), but
the multiplicity of (x∗, y∗) for that new system is not necessarily less than M/m. To fix the problem, based on the deflation lemma, we consider the following system instead:
ψ := F, G,∂F ∂Y , ∂G ∂Y ,· · · , ∂m−1F ∂Ym−1, ∂m−1G ∂Ym−1 .
The lemma then implies that (x∗, y∗) is a root of ψ of multiplicity n, with n ≤M/m. The invertibility assumption of ∂∂YmHm(x
∗, y∗) allows us to apply the implicit function
theorem to ∂∂Ym−m1−H1 at (x
∗, y∗). Replacing X by x∗+ξ, where ξ is a new variable, we can
find a power series J in K[[ξ]] and A inK[[ξ]][Y] such that
∂m−1H ∂Ym−1(x
∗
+ξ, Y) = (Y −J)A, J(0) =y∗ and A(0, y∗)6= 0.
Let us further replace Y by y∗ +ζ, where ζ is a new variable, and let us work in the power series ringK[[ξ, ζ]]. Since A(0, y∗) is nonzero,A(ξ, y∗+ζ) is a unit inK[[ξ, ζ]]. We deduce that in K[[ξ, ζ]], we have the equality
∂m−1H ∂Ym−1(x
∗
+ξ, y∗ +ζ) = ζ−(J −y∗).
This implies that in K[[ξ, ζ], (0,0) is a root of multiplicity n of the ideal generated by
∂αH ∂Yα(x ∗ +ξ, J) 0≤α<m−1 , ζ−(J−y∗), ∂αHc ∂Yα (x ∗ +ξ, J) 0≤α≤m−1 . Forα ≥0, define Hα := ∂αH ∂Yα(x ∗+ξ, J) and Hc α := ∂αHc ∂Yα (x ∗+ξ, J),
so that the above ideal is generated by
H0, H0c, H1, H1c, . . . , Hm−2, Hmc−2, ζ−(J−y
∗
), Hmc−1.
Remark that, with the exception of ζ−(J−y∗), all the above generators are in K[[ξ]]. Sinceζ−(J−y∗) has degree one inζ, we deduce that 0 is a root of multiplicityn of the ideal H0, H0c, H1, H1c, . . . , Hm−2, Hmc−2, H c m−1 ⊂K[[ξ]].
This proves in particular the following lemma.
Lemma 15. The integer n satisfies
n= min ({val(Hα)}0≤α<m−1 ∪ {val(Hαc)}0≤α<m),
where val denotes the ξ-adic valuation.
This allows us to to complete the definition of the signature σ(x∗, y∗): the last three components aren, the integera that realizes the minimum above (in case of a tie, choose the minimum), and a string K∈ {”H”,”Hc”} that indicates whether the minimum occurs forH orHc(in case of a tie, chooseH). Associated to stringK, we have the corresponding
polynomial K ∈ {F, G}, obviously defined sa K =H if K= ”H” and K =Hc otherwise.
The following lemma will help us give conditions on the preservation of the signature through specialization at primes, when for instance K=Q.
Lemma 16. Suppose that (x∗, y∗) has signature (m,H, n, a,K). Then (x∗, y∗) is a root of multiplicity n of h∂m−1H
∂Ym−1,
∂aK
∂Yai.
Proof. The proof amounts to going backward the previous derivation, but taking fewer
polynomials into account. By definition of a and K, and using Lemma 15, we see that n
is the ξ-adic valuation of
∂aK
∂Ya(x
∗
+ξ, J)
inK[[ξ]], that is, the multiplicity of 0 as a root of the equation ∂∂YaKa(x
∗+ξ, J). Equivalently,
it is the multiplicity of (0,0) as a root of the systemh∂aK
∂Ya(x
∗+ξ, J), ζ−(J−y∗)iin
K[[ξ, ζ]];
as we saw previously, this ideal coincides withh∂aK
∂Ya(x
∗+ξ, y∗+ζ),∂m−1H
∂Ym−1(x
∗+ξ, y∗+ζ)i.
Translating back the origin, this proves our claim.