A GLOBAL ŁOJASIEWICZ INEQUALITY

FOR ALGEBRAIC VARIETIES

by Shanyu Ji, János Kollár and Bernard Shiffman

Abstract. Let X be the locus of common zeros of polynomials f_{1}, …, f_{k} in n complex variables. A global upper bound for the distance to X is given in the form of a Łojasiewicz inequality. The exponent in this inequality is bounded by d^{min(n,k)} where d = max(3, degf_{i}). The estimates are also valid over an algebraically closed field of any characteristic.

AMS Subject Classification:Primary, 14M10, 32C99; Secondary, 32B99.

Let f be a real analytic function on ℝ^{n} and let Z = {x ∈ ℝ^{n}|f(x) = 0}. Let dist(x, Z) = inf_{z∈Z}||x − z|| where || · || denotes the Euclidean norm. For any x ∈ ℝ^{n} one expects to be able to compare dist(x, Z) and f(x). This is done by the Łojasiewicz inequality:

1. Theorem. (Łojasiewicz [L1, Theorem 17], see also [M,Theorem 4.1]) With the above notation, for any compact set K there are positive constants C and α such that

dist(x, Z)^{α} ≤ C · |f(x)| for every x ∈ K.

In general α can be large. For example [L2,p.85] if f(x, y) = y^{2m} + (y − x^{m})^{2} then α ≥ 2m^{2}. Also, it is not clear how α depends on f.

If Z ⊂ ℂ^{n} is defined by complex analytic equations f_{1} = ⋯ = f_{k} = 0 , then viewed as a real analytic set Z ⊂ ℂ^{n} ≅ ℝ^{2n} it is defined by f = 0 where f = |f_{1}|^{2} +⋯+|f_{k}|^{2} is a real analytic function. Thus the Łojasiewicz inequality applies to complex analytic or algebraic sets too.

If the defining equations f_{i} are polynomials, one would like to estimate the exponent α in terms of the degrees of the polynomials. Recently Brownawell [B1] (see also [BY,Section 3]) proved such a bound for polynomials over the complex field. A polynomial has more complex zeros than real ones; thus one expects the complex case to be easier. In fact Brownawell’s methods (and also ours) do not apply in the real case.

2. Theorem. (Brownawell [B1]) Let f_{1}, …, f_{k} ∈ ℂ[z_{1}, …, z_{n}] and let D = max deg f_{i}. Let Z = {z ∈ ℂ^{n}|f_{1}(z) = ⋯ = f_{k}(z) = 0}. Then there is a constant C > 0 such that

(\frac{min(dist(z, Z), 1)}{1 + ||z||^{2}})^{(n+1)^{2}D^{min(n,k)}} ≤ C · max_{i}|f_{i}(z)|,

where ||z||^{2} = |z_{1}|^{2} + ⋯ + |z_{n}|^{2}.

The aim of this paper is to find the best possible exponent in terms of the degrees of the polynomials f_{i}. We need the following notation:

3. Notation. Given natural numbers n ≥ 2 and d_{1} ≥ ... ≥ d_{k} let

B(n, d_{1}, ..., d_{k}) = \{ \begin{matrix}d_{1} · ... · d_{k} & if k ≤ n; \\ d_{1} · ... · d_{n−1} · d_{k} & if k > n. \end{matrix}

For technical reasons related to the proofs in [B2] and [K] we also define

B̄(n, d_{1}, ..., d_{k}) = (\frac{3}{2})^{j}B(n, d_{1}, ..., d_{k}) + θ,

where j = #{i < min(k, n) − 1|d_{i} = 2} and

θ = \{ \begin{matrix}1 if k > n and d_{n−1} = 2, \\ 0 otherwise. \end{matrix}

We extend the above notation to any sequence d_{1}, ..., d_{k} by first ordering it and then applying the above definitions.

4. Definition. Let K be an algebraically closed field. By an absolute value we mean a valuation | | : K → [0, ∞) which satisfies the triangle inequality (and which can be Archimedean or not). Any basis of K^{n} leads to a norm

||(x_{1}, …, x_{n})|| = \sqrt{|x_{1}|^{2} + ⋯ + |x_{n}|^{2}}.

If V ⊂ K^{n} then we define

dist(x, V) = inf_{y∈V} ||x − y||.

The following is our main result:

5. Theorem. (Łojasiewicz-type inequality)

Let K be an algebraically closed field (any characteristic) and let | | be an absolute value as in (4). Let f_{1}, …, f_{k} ∈ K[x_{1}, …, x_{n}] be polynomials and let d_{i} = degf_{i}. Assume that n ≥ 2. Let V = V (f_{1}, …, f_{k}) ⊂ K^{n} be the common zero set of these polynomials. Assume that V is nonempty.

Then there is a positive integer m ≤ B̄(n, d_{1}, ..., d_{k}) and a constant C > 0 (both depending on the f_{i}) such that

dist(x, V)^{m} ≤ C · max_{i}{|f_{i}(x)|} · (1 + ||x||)^{B̄(n,d_{1},...,d_{k})}

holds for all z ∈ K^{n}.

Since dist(z, Z) ≤ C′ · (1 + ||z||) holds for some C′ > 0, (5) implies the following improvement of Brownawell’s result:

6. Corollary. Let f_{1}, …, f_{k} ∈ ℂ[z_{1}, …, z_{n}] and let d_{i} = degf_{i}. Let Z ⊂ ℂ^{n} be the common zero set of these polynomials. Then there is a constant C > 0 such that

(6.1)  (\frac{min(dist(z, Z), 1)}{1 + ||z||})^{B̄(n,d_{1},...,d_{k})} ≤ C · max_{i}|f_{i}(z)| and

(6.2)  (\frac{dist(z, Z)}{1 + ||z||^{2}})^{B̄(n,d_{1},...,d_{k})} ≤ C · max_{i}|f_{i}(z)|

holds for all z ∈ ℂ^{n}. □

The proof of (5) will rest on Brownawell’s version [B2] of the effective Nullstellensatz [K]:

7. Theorem. [B2,Main Proposition] Let K be a field and let f̄_{1}, ..., f̄_{k} ∈ K[x_{0}, ..., x_{n}] be homogeneous polynomials of

degree d_{1}, …, d_{k} respectively. Assume that n ≥ 2.

Then there are prime ideals P_{1}, …, P_{s} containing (f̄_{1}, ..., f̄_{k}) and there are natural numbers e_{1}, …, e_{s} such that

∏_{i=1}^{s} P_{i}^{e_{i}} ⊂ (f̄_{1}, ..., f̄_{k}) and

∑_{i=1}^{s} e_{i} · degP_{i} ≤ B̄(n, d_{1}, ..., d_{k}).

The “analytic” part of the proof of (5) is based on the following Lemma in which we use the notation of (4)

8. Lemma. Let Z ⊂ K^{n} be an irreducible subvariety of dimension k and degree d. Then there are finitely many polynomials g_{i} of degree at most d vanishing on Z and a constant C such that

dist(x, Z)^{d} ≤ C · max_{i}{|g_{i}(x)|}.

Proof. Fix a generic projection Π : K^{n} → L where L is a k dimensional linear subspace. The restriction π : Z → L is finite of degree d and surjective. For any given x ∈ K^{n} let

dist_{Π}(x, Z) = min_{y∈π^{−1}(Π(x))}||x − y||.

It is clear that

dist(x, Z) ≤ dist_{Π}(x, Z).

Therefore it is sufficient to prove that

(9)  dist_{Π}(x, Z)^{d} ≤ C · max_{i}{|g_{i}(x)|}.

If dim Z = n − 1 then Z is defined by a single polynomial g ∈ K[x_{1}, …, x_{n}] of degree d. Since a different basis gives a norm which is bounded by constant multiples of the first norm from below and above, we may choose coordinates such that Π(x_{1}, …, x_{n}) = (x_{1}, …, x_{n−1}). Then g = cx_{n}^{d} + … where c ≠ 0. Let y_{1}, …, y_{d} ∈ Z be the preimages of Π(x) under π (with multiplicities). Then

dist(x, Z)^{d} ≤ ∏_{i=1}^{d} dist(x, y_{i}) = |c^{−1}g(x)| = |c^{−1}| · |g(x)|.

Thus C = |c^{−1}| is the desired constant.

We prove (9) by induction on the codimension. Suppose codim Z > 1; fix a hyperplane H ⊃ L and d + 1 generic lines A_{i} ⊂ kerΠ. Let p_{i} : K^{n} → H be the projections with kernel A_{i}. Let Z_{i} = p_{i}(Z) ⊂ H be the images. By assumption (9) is true for the Z_{i}.

It suffices to show that there is a constant C′ such that

dist_{Π}(x, Z) ≤ C′ · max_{i}{dist_{Π}(p_{i}(x), Z_{i})}.

To see this define the ϵ-neighborhood of A_{i} as

U_{ϵ}(A_{i}) =^{def} {x ∈ K^{n}|x = x_{a} + x_{h}; x_{a} ∈ A_{i}; x_{h} ∈ H and ||x_{h}|| < ϵ · ||x_{a}||}.

Note that if x ∉ U_{ϵ}(A_{i}) then ||x|| ≤ (1 + ϵ^{−1})||p_{i}(x)||. Now choose ϵ so that the ϵ-neighborhoods of the A_{i} do not intersect each other.

Let π^{−1}(Π(x)) = {y_{1}, …, y_{p}} (as sets); p ≤ d. There are (d + 1) lines A_{i} and at most d points x − y_{i}. Thus there is an index j (depending on x) such that

{x − y_{1}, …, x − y_{p}} ∩ U_{ϵ}(A_{j}) = ∅.

This implies that

dist_{Π}(x, Z) ≤ (1 + ϵ^{−1})max_{j}{dist_{Π}(p_{j}(x), Z_{j})}. □

10. Remark. It follows from the above proof that for a nonconstant polynomial f ∈ K[x_{1}, …, f_{n}]

dist(x, V (f))^{deg f} ≤ C · |f(x)|.

Now we can prove (5).

We introduce a new variable x_{0} and homogenize the polynomials f_{1}, ..., f_{k} to get f̄_{1}, ..., f̄_{k}. Let P_{1}, …, P_{s} be the prime ideals in (7). Assume that they are indexed such that x_{0} ∈ P_{r+1} ∩ ⋯ ∩ P_{s} and x_{0} ∉ P_{1} ∪ ⋯ ∪ P_{r}. Let Z_{1}, …, Z_{r} be the affine varieties in K^{n} corresponding to P_{1}, …, P_{r}. Then V = Z_{1} ∪ ⋯ ∪ Z_{r}.

By (8) for each Z_{i} we can find a finite collection of polynomials {g_{i,j}} of degree at most z_{i} := degZ_{i} = degP_{i} and a positive constant C_{i} such that g_{ij} vanishes on Z_{i} for each j and

dist(x, Z_{i})^{z_{i}} ≤ C_{i} · max_{j}{|g_{i,j}(x)|}.

Let e_{1}, …, e_{s} be as in (7). Then

(11)  \begin{matrix}dist(x, V)^{z_{1}e_{1}+...z_{r}e_{r}} ≤ \\ ∏_{i=1}^{r} dist(x, Z_{i})^{z_{i}e_{i}} ≤ \\ ∏_{i=1}^{r} C_{i}^{e_{i}} ∏_{i=1}^{r} max_{j}\{|g_{i,j}(x)|^{e_{i}}\} ≤ \\ C′ · max_{j_{1},...,j_{r}}\{| ∏_{i=1}^{r} g_{i,j_{i}}(x)^{e_{i}}|\}. \end{matrix}

Thus we need to understand the polynomials

∏_{i=1}^{r} g_{i,j_{i}}(x)^{e_{i}}.

By (7) we conclude that

(12)  x_{0}^{e_{r+1}+···+e_{s}} ∏_{i=1}^{r} g_{i,j_{i}}(x)^{e_{i}} ∈ (f̄_{1}, ..., f̄_{k}).

Since the degree of g_{i,j_{i}} is at most z_{i}, by (7) the degree of the left hand side in (12) is at most B̄(n, d_{1}, ..., d_{k}). Thus there are polynomials G_{i,j_{1},...,j_{r}} ∈ K[x_{1}, …, x_{n}] of degree at most B̄(n, d_{1}, ..., d_{k}) − d_{i} such that

∏_{i=1}^{r} g_{i,j_{i}}(x)^{e_{i}} = ∑_{i=1}^{k} G_{i,j_{1},...,j_{r}}f_{i}.

Note that if h ∈ K[x_{1}, …, x_{n}] has degree at most q then there is a constant C′′ such that

|h(x)| ≤ C′′ · (1 + ||x||)^{q}.

Thus for a suitable constant C′′

|G_{i,j_{1},...,j_{r}}(x)| ≤ C′′ · (1 + ||x||)^{B̄(n,d_{1},...,d_{k})−d_{i}},

where C′′ is independent of i, j_{1}, …, j_{r}. Therefore by (11)

(13)  \begin{matrix}dist(x, V)^{z_{1}e_{1}+...z_{r}e_{r}} ≤ \\ C′ · max_{j_{1},...,j_{r}}\{| ∑_{i} G_{i,j_{1},...,j_{r}}f_{i}|\} ≤ \\ kC′C′′ · max_{i}\{|f_{i}(x)|\} · (1 + ||x||)^{B̄(n,d_{1},...,d_{k})}. \end{matrix}

Take m = ∑_{i=1}^{r} z_{i}e_{i} and C = kC′C′′ to get (5). □

Note that we proved in fact a slightly stronger statement:

(14)  dist(x, V)^{m} ≤ C · max_{i}{|f_{i}(x)| · (1 + ||x||)^{B̄(n,d_{1},...,d_{k})−d_{i}}}.

In this form both the bound on m and the exponents of 1+||x|| are the best possible, provided that d_{i} ≠ 2.

15. Example. This variant of an example given by Masser and Philippon (see [B1;K,2.3]) shows that the upper bound on m in (14) is sharp (for d_{i} ≠ 2).

Let f_{1} = x_{2} − x_{1}^{d_{1}}, f_{2} = x_{3} − x_{2}^{d_{2}}, …, f_{n−1} = x_{n} − x_{n−1}^{d_{n−1}}, f_{n} = x_{n}^{d_{n}}. Then V (f_{1}, …, f_{n}) = {0}. Let

x(t) = (t, t^{d_{1}}, t^{d_{1}d_{2}}, …, t^{d_{1}d_{2}...d_{n−1}}).

Then dist(x(t), 0) ≈ |t| for small |t| but

max|f_{i}(x(t))| = |f_{n}(x(t))| = |t|^{d_{1}d_{2}...d_{n}}.

16. Example. This example shows that in some cases the only value of m that works in (5) is m = 1.

In K[x, y] let f_{1} = y, f_{2} = y(x − 1)^{s} − x where s ≥ 2. Then B̄(2, 1, s + 1) = s + 1 and V (f_{1}, f_{2}) = {0}. Consider the family of points z(t) = (t, t(t − 1)^{−s}). Then f_{2}(z(t)) = 0 and for large values of |t| we have

dist(z(t), {0}) = ||z(t)|| ≈ |t| and |f_{1}(z(t))| ≈ |t|^{1−s}.

Thus

max_{i=1,2}{|f_{i}(z(t))| · (1 + ||z(t)||)^{B̄(n,d_{1},d_{2})−d_{i}}} = |f_{1}(z(t))| · (1 + ||z(t)||)^{s} ≈ |t|.

Hence we must take m = 1 in (5).

This example also shows that the exponent 2 of ||z|| in (6.2) cannot be made smaller if the degrees go to infinity.

17. Remark. One can interpret (5) as follows: Given a system of equations f_{1} = ⋯ = f_{k} = 0 over an algebraically closed field K, let (x_{1}, …, x_{n}) be an approximate solution. Then there is an actual solution near (x_{1}, …, x_{n}). From this point of view the assumption that V be nonempty is very inconvenient. This form is especially interesting when the absolute value is non-archimedean, e.g. when K is the algebraic closure of a complete discrete valuation ring. However in this case one would like to prove a similar result without assuming that K is algebraically closed, or even for equations over any complete local ring. Such results are known [A,Chapter 6] but the bounds are probably far from being optimal.

18. Remark. Let us take this opportunity to correct an error in [K]. In the formulation of [K,Prop. 1.10] the exponent should be the above B(n, d_{1}, ..., d_{k}) instead of N(n, d_{1}, ..., d_{k}). (See [K, Section 4].) These two functions are conjecturally equal but the equality is proved only if all the d_{i} are different from 2 ([K,1.9]).

Acknowledgement. We wish to thank C. Berenstein, Y. Shimizu, A. Yger and S. Zucker for their helpful conversations about this work.

Partial financial support for the first author was provided by a University of Houston Research Initiation Grant. Partial financial support for the second author was provided by the NSF under grant numbers DMS-8707320 and DMS-8946082 and by an A. P. Sloan Research Fellowship. Partial financial support for the third author was provided by the NSF under grant numbers DMS-8701808 and DMS-8901571.

References

[A] M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math. IHES 36 (1969), 23-58.

[BY] C. A. Berenstein and A. Yger, Bounds for the degrees in the division problem, Michigan Math. J. (to appear).

[B1] W. D. Brownawell, Local diophantine Nullstellen inequalities, Journal AMS 1 (1988), 311-322.

[B2] W. D. Brownawell, A prime power product version of the Nullstellensatz, Michigan Math. J. (to appear).

[K] J. Kollár, Sharp effective Nullstellensatz, Journal AMS 1 (1988), 963-975.

[L1] S. Łojasiewicz, Sur le problème de la division, Studia Math 18 (1959), 87-136.

[L2] S. Łojasiewicz, Ensembles Semi-analytiques, IHES, Bures-sur-Yvette, 1965.

[M] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966.

Department of Mathematics University of Houston Houston,TX 77204

Department of Mathematics University of Utah Salt Lake City, UT 84112

Department of Mathematics The Johns Hopkins University Baltimore, MD 21218