Bounds on Distance to Variety in Terms of Coefficients of Bivariate Polynomials

0.08 MB·

by  Vikram Sharma

Bounds on Distance to Variety in Terms of
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Coefficients of Bivariate Polynomials
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0
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Vikram Sharma
n
The Institute of Mathematical Sciences, HBNI
a
J Chennai, India 600113

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] Let f C[x] be a univariate polynomial of degree d with roots α1,...,αd.
V For a poin∈t z C, let sep(f,z) := min z α . Then for all points z C we
i i
∈ | − | ∈
C know that the logarithmic derivative at z is
.
h
t f′(z) d 1
a = . (1)
f(z) z α
m i
i=1 | − |
X
[
and more generally for any k 1 we have
≥
1
v f(k)(z) 1
= . (2)
3
f(z) (z α )(z α )...(z α )
1 1≤i1<i2X<···<ik≤d − i1 − i2 − ik
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8 Takingabsolutevalueonbothsides,applyingtriangularinequalityontheRHS,
0 and observing that the number of terms on the RHS is d and each is smaller
k
1. than 1/sep(f,z) we get the following bound: for k 1
≥ (cid:0) (cid:1)
0
7 f(k)(z) 1/k d
1 . (3)
f(z) ≤ sep(f,z)
: (cid:12) (cid:12)
v (cid:12) (cid:12)
i Another way to interpret th(cid:12)is bound(cid:12) is to state it as follows:
X (cid:12) (cid:12)
r f(z)
a sep(f,z) d min . (4)
≤ 1≤k≤d f(k)(z)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Similar bounds are also derived in [Hen74, p(cid:12). 452]. (cid:12)In this short note, we will
generalize this result to bivariate polynomial f(x,y) C[x,y]. The analogue
∈
result will have the following form: the left hand side will be the distance of a
point p from the variety of f, and the RHS will consist of the total degree of f
anda quantitydependent onthe absolutevaluesoff andits partialderivatives
evaluated at p. We first establish some notation. For k 0, define
≥
∂kf(p)
f (p):= . (5)
i,k ∂ix∂k−iy
1
Let D be the total degree of f, V(f) C2 be the variety of f, and
⊆
sep(p,V(f)):= inf p x (6)
x∈V(f)k − k
be the distance function to V(f).
Theideaforderivingtheboundisasfollows. Considerapointp=(p ,p )
x y
C2 V(f). In order to derive an upper bound on sep(p,V(f)), we will conside∈r
\
all the lines through p. These lines intersect the curve f(x,y) = 0 at finitely
many points that can be obtained as roots of a univariate polynomial. For in-
stance,considerthe intersectionoftheline x=p withthe curvef =0. Apply
x
the upper bound in (4) to the resulting univariate polynomial we obtain that
f(p) 1/k
sep(p,V(f)) D min .
≤ 1≤k≤D(cid:12)f0,k(p)(cid:12)
(cid:12) (cid:12)
Similarly, considering the intersection of the (cid:12)(cid:12)line y =(cid:12)(cid:12)py with the curve f = 0
we also get that
D f(p) 1/k
sep(p,V(f)) min k! .
≤1≤k≤D(cid:12) (cid:18)k(cid:19)fk,0(p)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
How do we get the terms corresponding(cid:12)to the mixed p(cid:12)artial derivatives? We
consider all the lines with slope tanθ, as θ varies from 0 to 2π, and take the
minimum of the absolute value of the correspondingroots over all θ. Since this
functionisperiodic inθ, itmakessenseto usesometoolsfromFourieranalysis.
The remaining section develops this idea into full detail.
Considering f as a polynomial in x with coefficients in C[y], from the local
parameterization of algebraic curves [Wal78], we know that in a certain neigh-
borhood of a point (x,y) C2 V(f) we can express
∈ \
d(y)
f(x,y):=K (x α (y)), (7)
i
−
i=1
Y
where α ’s are holomorphic functions of y, the degree d(y) deg(f,x) depends
i
on the y-coordinate, and K C is some constant. Differe≤ntiating both sides
∈
with respect to x and factoring f(x,y) from the RHS we obtain that
d(y)
1
f (x,y)=f(x,y) , (8)
1,0
x α (y)
i
i=1 −
X
and in general
1
f (x,y)=f(x,y) .
k,0
(x α (y))...(x α (y))
1≤i1<i2<X···<ik≤d(y) − i1 − ik
2
Following the argument used to derive (3) in the univariate setting, we obtain
that for any point p C2
∈
f (p) 1/k D
k,0
, (9)
f(p) ≤ sep(p,V(f))
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Note that if there is an as(cid:12)ymptote(cid:12) at y then d(y)< deg(f,x); also, if d(y) =0
then the bound above on the partial derivatives trivially holds since all the
partial derivatives vanish.
Wewanttoderiveasimilarboundforthemixedpartialderivativesf (p).
i,k−i
Toobtainthis,wechangethecoordinatesystemandthenconsidertheintersec-
tion with either the horizontal or vertical axis. Consider the following change
of coordinates:
x 1 eθ e−ψ X X
:= =U
y √2 e−ψ e−θ Y · Y
(cid:20) (cid:21) (cid:20) − (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
where =√ 1 and θ,ψ are any angles; we will later set ψ =0. Note that the
−
matrix U is unitary since
1 eθ e−ψ 1 e−θ eψ 1 0
UU† = = .
√2 eψ e−θ · √2 e−ψ eθ 0 1
(cid:20) − (cid:21) (cid:20) − (cid:21) (cid:20) (cid:21)
Define
eθX +e−ψY eψX e−θY
F(X,Y):=f(U(X,Y))=f , − .
√2 √2
(cid:18) (cid:19)
By repeated applications of the chain rule of partial differentiation we know
that
k i k−i
k ∂x ∂y
F (X,Y)= (f U(X,Y)) .
k,0 i,k−i
i ◦ ∂X ∂X
i=0(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
Observe that
∂x/∂X =eθ/√2, ∂y/∂X =eψ/√2.
Thus,
k
k
F (X,Y)= (f U(X,Y))eiθ3(k−i)ψ2−k. (10)
k,0 i,k−i
i ◦
i=0(cid:18) (cid:19)
X
Since total degree of F is the same as the total degree of f, it follows from (9)
that for a point p C2
∈
F (U−1(p)) 1/k D D D
k,0
= = ,
F(U−1(p)) ≤ sep(U−1(p),V(F)) sep(U−1(p),U−1(V(f))) sep(p,V(f))
(cid:12) (cid:12)
(cid:12) (cid:12) (11)
(cid:12) (cid:12)
w(cid:12)here the last s(cid:12)tep follows from the fact that U is a unitary transformation.
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Moreover, as F(U−1(p)) = f(p), from (10) and (11) we obtain that for all
choices of θ [ π,π]
∈ −
1/k
k
k f (p) D
i,k−i eiθe(k−i)ψ2−k . (12)
(cid:12) i f(p) (cid:12) ≤ sep(p,V(f))
(cid:12)Xi=0(cid:18) (cid:19) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Let P(θ) be t(cid:12)he function inside the absolute v(cid:12)alue on the LHS above. Since it
is a Fourier series in θ, from Parseval’s theorem we know that
k k f (p) 2 1 π
i,k−i e(k−i)ψ2−k = P(θ)2dθ.
i f(p) 2π | |
i=0(cid:18)(cid:18) (cid:19)(cid:12) (cid:12) (cid:19) Z−π
X (cid:12) (cid:12)
(cid:12) (cid:12)
Substituting the uppe(cid:12)r bound (1(cid:12)2) on P(θ) in the integral on the RHS above
| |
we further obtain that
k k f (p) 2 D 2k
i,k−i e(k−i)ψ2−k . (13)
i f(p) ≤ sep(p,V(f))
i=0(cid:18)(cid:18) (cid:19)(cid:12) (cid:12) (cid:19) (cid:18) (cid:19)
X (cid:12) (cid:12)
(cid:12) (cid:12)
Choosing ψ =0, we o(cid:12)btain (cid:12)
k k f (p) 2 f (p) 2
i,k−i e(k−i)ψ2−k > max j,k−j 2−k
i f(p) j=0,...,k f(p)
i=0(cid:18)(cid:18) (cid:19)(cid:12) (cid:12) (cid:19) (cid:18)(cid:12) (cid:12) (cid:19)
X (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Combining this w(cid:12)ith (13) w(cid:12)e obtain the following: (cid:12) (cid:12)
Theorem 1. For a point p C2 that is not a zero of f
∈
f (p) 1/k 2D
i,k−i
max < . (14)
i=0,...,k f(p) sep(p,V(f))
(cid:12) (cid:12)
(cid:12) (cid:12)
The bound above can al(cid:12)so be inte(cid:12)rpreted as an upper bound on the sep-
(cid:12) (cid:12)
aration of a point p from the variety V(f) in terms of the coefficients of the
polynomial. Canaconverseboundbegiven,i.e.,alowerboundontheseparation
in terms of the coefficients. We next derive such a bound.
Suppose f(0)=0, then we want to derive a lower bound on sep(0,V(f)) in
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terms of the coefficients. Clearly, any x=(x,y) for which
k
a > a xi y k−i (15)
0,0 i,k−i
| | | || | | |
k≥1i=0
XX
cannot be on the variety of f. Define
1/k
k! a
i,k−i
γ := max max ,
1≤k≤D0≤i≤k ki (cid:12) a0 (cid:12)!
(cid:12) (cid:12)
(cid:0) (cid:1)(cid:12) (cid:12)
(cid:12) (cid:12)
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where D is the total degree. Then it follows that (15) is equivalent to
k
γ k γ
1> xi y k−i = (x + y )k >exp(γ x ) 1.
1
k! i | | | | k! | | | | k k −
k≥1 i=0(cid:18) (cid:19) k≥1
X X X
Therefore, if x is such that x γ < ln2 then f(x) > 0. In general, for any
1
point p C2 we can applykthke argument abov|e to|the shifted polynomial to
∈
obtain the following: if
f (p) 1/k
γ (p):= max max i,k−i , (16)
f 1≤k≤D0≤i≤k f(p)
(cid:12) (cid:12)
(cid:12) (cid:12)
then (cid:12) (cid:12)
ln2(cid:12) 1(cid:12)
sep(p,V(f)) . (17)
≥ √2γ(p) ≥ 3γ(p)
Besidestheirintrinsicinterest,suchboundsareusefulinanalyzingthe com-
plexity of certain algorithms. For instance, the bound given in (3) has been
useful in bounding the running time of certain root isolation algorithms using
the continuous amortization framework [Bur16, SB15]. We expect the general-
izationgivenabove to be useful in deriving similar bounds onthe running time
of generalizations of corresponding algorithms that generally use subdivision
(e.g., [PV04]).
Acknowledgement: The author is grateful to Chee Yap and Bernard
Mourrain for their feedback on earlier drafts of the results presented here.
References
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[SB15] Vikram Sharma and Prashant Batra. Near optimal subdivision algo-
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