On Mahler’s measure for polynomials in several variables

Imre Z. Ruzsa*

Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Pf. 127, H–1364 Hungary. e-mail: ruzsa@math-inst.hu

To Professor Andrzej Schinzel, on the occasion of his sixtieth birthday.

Abstract. An 1915 theorem of Szegő expresses the Mahler measure of a polynomial in terms of L_{2} norms of its multiples. We extend this to polynomials in several variables.

1. Introduction

Let f be a nonzero polynomial. Its Mahler measure is defined as

M(f) = exp ∫_{0}^{1} log |f(e(t))| dt

where e(t) = e^{2πit}. The L_{p} norm of f will be meant on the same set:

∥f∥_{p} = (∫_{0}^{1} |f(e(t))|^{p} dt)^{1/p} ,

for 0 < p < ∞. ∥f∥_{∞} will denote the maximum norm on the same set.

A theorem of Szegő [2] asserts that

M(f) = inf_{g(0)=1} ∥fg∥_{2}.  (1.1)

We investigate the possibility to extend this result to polynomials in several variables.

To facilitate the expression we introduce the following vectorial notation. We set t = (t_{1}, ..., t_{n}) and similarly for other letters; the origin will be 0 = (0, ..., 0). We put

e(t) = (e(t_{1}), ..., e(t_{n})),

* Supported by Hungarian National Foundation for Scientific Research, Grant No. T 017433 and we shall abbreviate ∫ ... ∫ f dt_{1}... dt_{n} as ∫ f dt. A general monomial will be denoted as x^{k} = x_{1}^{k_{1}}...x_{n}^{k_{n}}. xy is the usual scalar product ∑ x_{j}y_{j}.

With this notation the definition of Mahler measure and L_{p} norm for polynomials in n variables takes the form

M(f) = exp ∫_{[0,1)^{n}} log |f(e(t))| dt

and

∥f∥_{p} = (∫_{[0,1)^{n}} |f(e(t))|^{p} dt)^{1/p} ,

for 0 < p < ∞. ∥f∥_{∞} will denote the maximum norm on the same set.

An immediate generalization of (1.1) would assert

M(f) = inf_{g(0)=1} ∥fg∥_{2}.  (1.2)

We show that (1.2) fails for n = 2 for the polynomial x − y.

Definition 1.1. For 0 < p ≤ ∞ we define the Szegő p-measure of a polynomial f by the formula

S_{p}(f) = inf_{g(0)=1} ∥fg∥_{p}.

Statement 1.2.

For the polynomial f_{0}(x, y) = x − y we have M(f_{0}) = 1 < S_{2}(f_{0}) = \sqrt{2}.

Proof. Any polynomial f_{0}g where the constant term of g is 1 will contain the terms x and y. Hence, writing f_{0}g = ∑ a_{kl}x^{k}y^{l}, an application of Parseval’s formula yields

∥f_{0}g∥_{2}^{2} = ∑ |a_{kl}|^{2} ≥ 2.

The choice g = 1 shows that we can have equality here.

To calculate M(f_{0}) we use Jensen’s identity

M(f) = |f(0)| ∏ max(1, |α_{j}|^{−1})  (1.3)

where f is a polynomial in one variable and the α_{j}’s are the roots of f. By applying this to the polynomials f(x) = f_{0}(x, y) for each fixed value of y, |y| = 1 we see that in the double integral defining log M(f_{0}) each inner integral vanishes.

In the sequel we find a generalization of Szegő’s theorem by appropriately extending the class g(0) = 1, and describe those polynomials for which equality (1.2) stays in power.

For the case of one variable, Theorems 1, 2, 3 all reduce to Szegő’s theorem, so as a byproduct we obtain three new proofs of this result, of which the third seems to be the natural one.

Acknowledgement. I am very much indebted to Professor Andrzej Schinzel. I heard about this problem from a talk he gave in Debrecen. He kindly gave me further information, which included showing me the manuscript of his book in preparation. I profited much from his remarks and corrections to the previous versions of this paper. Theorem 1 below was also first conjectured by him, in one of the very informative letters he sent me on this subject.

2. Mahler measure and extremal terms

Let g(x) = ∑ a_{k}x^{k} be a polynomial in n variables. We say that a_{k}x^{k} is an extremal term of g if a_{k} ≠ 0 and the point k is not in the convex hull of the other exponent vectors corresponding to nonzero coefficients. Write

E(g) = max{|a_{k}| : a_{k}x^{k} is an extremal term of g}.

One of the extremal terms is the lexicographically first nonzero term, that is, the one for which k_{1} is minimal, from the vectors minimalizing k_{1} the value of k_{2} is minimal, and so on. Put

L(g) = {|a_{k}| : a_{k}x^{k} is the lexicographically first nonzero term of g}.

Clearly we have E(g) ≥ L(g).

Theorem 1.

For every 0 < p ≤ ∞ we have

M(f) = inf_{E(g)=1} ∥fg∥_{p} = inf_{L(g)=1} ∥fg∥_{p}.

For a polynomial in one variable if f(x) = x^{k}f_{1}(x) with f_{1}(0) ≠ 0, then L(f) = |f_{1}(0)|. Since a multiplication by x^{k} does not affect the value of M or ∥...∥_{p}, Theorem 1 reduces to (a slight generalization of) Szegő’s theorem.

Lemma 2.1. For every nonzero polynomial g we have

|g(0)| ≤ L(g) ≤ E(g) ≤ M(g).

Proof. The last inequality is a result of Smyth [1] , the first two are obvious.

Proof of Theorem 1, upper estimate. We show that

M(f) ≤ inf_{E(g)=1} ∥fg∥_{p} ≤ inf_{L(g)=1} ∥fg∥_{p}.

Indeed, if E(g) = 1, then by the previous lemma we have M(g) ≥ 1, hence

∥fg∥_{p} ≥ M(fg) = M(f)M(g) ≥ M(f).

This shows the first inequality; the second is obvious.

The lower estimate will be based on a series of lemmas.

Lemma 2.2. If r < 1 and |z| ≤ 1, z ≠ 1, then

|\frac{1 − rz}{1 − z} | ≥ \frac{1 + r}{2} .

Proof. We have

\frac{1 − rz}{1 − z} = \frac{1 + r}{2} + \frac{1 − r}{2} \frac{1 + z}{1 − z}

and

Re \frac{1 + z}{1 − z} = \frac{1 − |z|^{2}}{|1 − z|^{2}} ≥ 0,

hence

|\frac{1 − rz}{1 − z} | ≥ Re \frac{1 − rz}{1 − z} ≥ \frac{1 + r}{2} .

Lemma 2.3.

Let f be a polynomial of degree m (in one variable),

ν_{k} = ∫_{0}^{1} log |f(e(t))|e(−kt) dt,

0 < r < 1 and

G(x) = ∑_{k=−∞}^{∞} ν_{k}r^{|k|}x^{k}.

For |x| = 1 and f(x) ≠ 0 we have

G(x) ≥ log |f(x)| + m log \frac{1 + r}{2} .  (2.1)

Proof. Consider first the case when f(x) = 1 − αx. We are going to calculate G(x). Assume first that |α| < 1. Applying the power series expansion of log(1 − z) we

obtain

log(1 − αx) = ∑_{k=1}^{∞} \frac{1}{k} α^{k}x^{k},

hence (recall that |x| = 1, so \overline{x} = x^{−1})

log |1 − αx| = Re log(1 − αx) = ∑_{k=1}^{∞} \frac{1}{2k} (α^{k}x^{k} + \overline{α}^{k}x^{−k}) .  (2.2)

This shows that

ν_{k} = \{ \begin{matrix}\frac{1}{2k} α^{k} & for k > 0, \\ 0 & for k = 0, \\ \frac{1}{2|k|} \overline{α}^{|k|} & for k < 0. \end{matrix}  (2.3)

To prove (2.3) for |α| = 1 we first show that the functions log |1 − rαx| tend to log |1 − αx| in L_{1}-norm as r → 1. With the notation

ϕ(t) = log |\frac{1 − αe(t)}{1 − rαe(t)}| ,

this amounts to

∫ |ϕ(t)| dt → 0

as r → 1. We know ∫ ϕ(t) = 0 by Jensen’s formula and ϕ(t) ≤ log 2/(1 + r) for each t by Lemma 2.2. We introduce the notation

t_{+} = max(t, 0).

We obtain

∫ |ϕ(t)| dt = 2 ∫ ϕ(t)_{+} dt ≤ 2 log \frac{2}{1 + r}

as claimed. Consequenly we can find the value of ν_{k} by applying (2.3) for rα in the place of α and taking the limit for r → 1.

Thus for |α| ≤ 1 the value of ν_{k} is given by (2.3) and we have

G(x) = ∑_{k=1}^{∞} \frac{1}{2k} r^{k}(α^{k}x^{k} + \overline{α}^{k}x^{−k}) = log |1 − rαx|

where in the last step we used (2.2) for rα in the place of α again. Hence (2.1) follows from the previous lemma, applied to the number z = αx.

Now assume |α| > 1. In this case we have 1 − αx = −αx(1 − (αx)^{−1}), hence

log |1 − αx| = log |α| + log |1 − \frac{1}{αx}| .

Thus a similar calculation yields

log |1 − αx| = log |α| + ∑_{k=1}^{∞} \frac{1}{2k} (α^{−k}x^{−k} + \overline{α}^{−k}x^{k}) ,

then

ν_{k} = \{ \begin{matrix}\frac{1}{2k} \overline{α}^{−k} & for k > 0, \\ log |α| & for k = 0, \\ \frac{1}{2|k|}α^{−|k|} & for k < 0, \end{matrix}

finally

G(x) = log |α| + log |1 − \frac{rx}{\overline{α}}| = log |r − αx|.

Hence (2.1) follows from the previous lemma, applied to the number z = (αx)^{−1}. The general case follows by decomposing a general polynomial as

f(x) = cx^{l} ∏_{j=1}^{m−l}(1 − α_{j}x),

applying the inequality to each factor 1 − α_{j}x and summing the resulting inequalities.

Remark 2.4. By substituting the definition of ν_{k} and exchanging the integration and summation we obtain that

G(x) = ∫_{0}^{1} log |f(e(t))|\frac{1 − r^{2}}{1 + r^{2} − 2r Re xe(−t)} dt.  (2.4)

Now we consider polynomials in several variables. Define

ν_{k}(f) = ∫_{[0,1)^{n}} e(−kt) log|f(e(t))| dt.  (2.5)

Observe that

ν_{0}(f) = log M(f).

Lemma 2.5.

Let f be a polynomial of degree m_{1}, ..., m_{n} in x_{1}, ..., x_{n}. Let 0 < r_{j} < 1 and

G(x) = ∑_{k_{1},...,k_{n}=−∞}^{∞} ν_{k}(f)r_{1}^{|k_{1}|}...r_{n}^{|k_{n}|}x^{k}.

For |x_{1}| = ... = |x_{n}| = 1 and f(x) ≠ 0 we have

G(x) ≥ log |f(x)| + ∑ m_{j} log \frac{1 + r_{j}}{2} .  (2.6)

Proof. By substituting the definition of ν_{k} and exchanging the integration and summation we obtain that

G(x) = ∫_{[0,1)^{n}} log |f(e(t))|\frac{1 − r_{1}^{2}}{1 + r_{1}^{2} − 2r_{1} Re x_{1}e(−t_{1})}...\frac{1 − r_{n}^{2}}{1 + r_{n}^{2} − 2r_{n} Re x_{n}e(−t_{n})} dt.

For any fixed value of t_{2}, ..., t_{n} we have

∫_{0}^{1} log |f(e(t))|\frac{1 − r_{1}^{2}}{1 + r_{1}^{2} − 2r_{1} Re xe(−t_{1})} dt_{1} ≥ log|f(x, e(t_{2}), ..., e(t_{n}))| + m_{1} log \frac{1 + r_{1}}{2}

by applying the previous Lemma and Remark for the polynomial

f_{t_{2},...,t_{n}}(x) = f(x, e(t_{2}), ..., e(t_{n})).

Now we multiply this inequality by

\frac{1 − r_{2}^{2}}{1 + r_{2}^{2} − 2r_{2} Re x_{2}e(−t_{2})},  (2.7)

integrate and apply the previous lemma again. (We used the fact that the kernel function in (2.7) is nonnegative and its integral is 1.) Repeating this procedure altogether n − 1 times we arrive at the desired inequality.

Proof of Theorem 1, lower estimate.

By virtue of the already already settled part and the monotonicity of ∥...∥_{p} in p, to complete the proof of Theorem 1 it is sufficient to show the following:

For every nonzero polynomial f in n variables and for every ε > 0 there is another polynomial g such that L(g) = 1 and

∥fg∥_{∞} < (1 + ε)M(f).

Take two real numbers 0 < r < 1 and δ > 0.

Let I be the collection of those vectors k ≠ 0 of integers whose first nonzero coordinate is positive, and consider the sum

H(x) = 2 ∑_{k∈I} ν_{k}r^{|k_{1}|+...+|k_{n}|}x^{k}.

From now on we will always assume that |x_{j}| = 1 for all j. Since for every k ≠ 0 exactly one of the relations k ∈ I and −k ∈ I is true, we have (with the G of the previous lemma)

Re H(x) + ν_{0} = G(x).

Thus by Lemma 2.5 we have with m = m_{1} + ... + m_{n}

log |f(x)| ≤ log G(x) − m log \frac{1 + r}{2} = Re H(x) + ν_{0} − m log \frac{1 + r}{2} ,

that is (recall that ν_{0} = log M(f))

|f(x)| ≤ (\frac{2}{r + 1})^{m} M(f) |e^{H(x)}| .  (2.8)

Since the numbers ν_{k} are bounded, the series defining H is uniformly convergent and we can find a finite subsum, say H_{1}, such that

|H_{1}(x) − H(x)| ≤ log(1 + δ)

for all x. Consequently

|e^{−H_{1}(x)}| ≤ (1 + δ) |e^{−H(x)}| .

Since H_{1} is bounded, we can find an integer s such that the function

g_{1}(x) = ∑_{j=0}^{s} \frac{(−1)^{j}}{j!} H_{1}(x)^{j}

satisfies

|g_{1}(x)| ≤ (1 + δ) |e^{−H_{1}(x)}| ≤ (1 + δ)^{2} |e^{−H(x)}|

for all x.

Now H_{1} is a finite sum of terms of the form ax^{k} with k ∈ I. Since I is closed under addition, each H_{1}^{j} for j ≥ 1 is a finite sum of terms of the same kind. Consequently g_{1} is of the same form, except for the summand 1 coming from j = 0.

Now for a suitable positive integer q

g(x) = (x_{1}...x_{n})^{q}g_{1}(x)

is a polynomial with L(g) = 1, namely its lexicographically first term will be x_{1}^{q}...x_{n}^{q}, and for |x_{j}| = 1 we have

|g(x)| = |g_{1}(x)| ≤ (1 + δ)^{2} |e^{−H(x)}| .

Multiplying this by (2.8) we obtain

|f(x)g(x)| ≤ (1 + δ)^{2} (\frac{2}{r + 1})^{m} M(f) < (1 + ε)M(f)

if δ and r were suitably chosen.

3. The Szegő class and Fourier coefficients

We show that the possibility to extend Szegő’s original formulation, that is, the validity of (1.2), can also be described in terms of the Fourier coefficients ν_{k}(f).

Definition 3.1. A vector k = (k_{1}, ..., k_{n}) is mixed, if there are both positive and negative values among the coordinates k_{j}.

Theorem 2.

a) If there is a mixed k such that ν_{k}(f) ≠ 0, then

S_{p}(f) > M(f)

for every 0 < p ≤ ∞.

b) If ν_{k} = 0 for all mixed k, then

S_{p}(f) = M(f)  (3.1)

for all 0 < p ≤ ∞.

Definition 3.2. We say that f is in the Szegő class, if it satisfies (3.1).

Proof of Theorem 2, part b). It is sufficient to consider the case p = ∞.

Let f be a polynomial of degree n such that ν_{k} = 0 whenever k is mixed. Select a number 0 < r < 1 and consider the functions G and H from the proof of Theorem 1. Recall that for |x_{j}| = 1 we have

Re H(x) + ν_{0} = G(x).  (3.2)

The novelty is that, because of the lack of mixed exponents, every vector in I has only nonnegative coordinates. H is therefore defined by a power series, and (since ν_{k} is bounded) it will be analytic in the domain |x_{j}| < 1/r. Hence for any δ > 0 we can find a polynomial h such that

|h(x) − exp(−H(x))| < δ

for all |x_{j}| ≤ 1. Since H(0) = 0, for the polynomial

g(x) = h(x)/h(0)

this implies g(0) = 1 and

|g(x)| < (1 + ε)| exp −H(x)|

if δ is so small that

δ < (ε − δ − εδ) exp − max_{|x_{j}|≤1} |H(x)|.

By (3.2) this means

|g(x)| < (1 + ε) exp(ν_{0} − G(x)) = (1 + ε)M(f) exp −G(x).

Hence for |x_{j}| = 1 we have

|f(x)g(x)| ≤ (1 + ε)M(f) exp(log |f(x)| − G(x)) ≤ (1 + ε) (\frac{2}{r + 1})^{m} M(f)

by Lemma 2.5. Since ε can be arbitrarily small and r can be arbitrarily close to 1, the proof is completed.

To prove part a) we need some preparation.

Lemma 3.3.

Let f be a polynomial in n variables such that f(0) = 1 and let k be a mixed vector. We have

|ν_{k}(f)| ≤ ν_{0}(f).

Proof. Since k is mixed, we can find a vector a_{1} with strictly positive integer coordinates such that ka_{1} = 0. Given this vector a_{1}, let a_{2}, ..., a_{n} be arbitrary vectors with positive integer coordinates that together with a_{1} form a basis of the space. The matrix A, obtained by writing the vectors a_{1}, ..., a_{n} as rows, will have the following properties: det A ≠ 0 and

Ak = m = (0, m_{2}, ..., m_{n}).

Consider the polynomial

g(x) = f(x^{a_{1}}, ..., x^{a_{n}}).

Since each entry of the matrix is positive, g is of the form

g(x) = 1 + x_{1}...x_{n}h(x),  (3.3)

hence g(x) = 1 if x_{j} = 0 for at least one j. Furthermore we have

g(e(t)) = f(e(At)).

From the unicity of the Haar measure it follows that for every function F we have

∫_{[0,1)^{n}} F (e(t)) dt = c ∫_{[0,1)^{n}} F (e(At)) dt

with a positive absolute constant c. On applying this for the function identically 1 we see that c = 1. In particular, we have

ν_{Ab}(g) = ν_{b}(f)

for every integral vector b. The choice b = k yields ν_{k}(f) = ν_{m}(g).

Now we have

ν_{m}(g) = ∫ e(−m_{2}t_{2} − ... − m_{n}t_{n}) log |g(e(t))| dt

so

|ν_{m}(g)| ≤ ∫ ... ∫ |∫ log |g(e(t))| dt_{1}| dt_{2}... dt_{n}.  (3.4)

Now for any fixed value of t_{2}, ..., t_{n} we have
∫ log |g(e(t))| dt_{1} ≥ log |g(0, e(t_{2}), ..., e(t_{n})) = 0.

Here the inequality follows from Jensen’s identity (1.3), and the equality from property (3.3) of the polynomial g. Thus (3.4) reduces to

|ν_{m}(g)| ≤ ∫ ... ∫ ∫ log |g(e(t))| dt_{1} dt_{2}... dt_{n} = ν_{0}(g) = ν_{0}(f).

Write

Δ(f) = ∫ |log |f| − ν_{0}(f)| dt.

Lemma 3.4.

For every p ∈ (0, ∞) we have

Δ(f) ≤ \frac{\sqrt{8}}{p} (∥\frac{f}{M(f)}∥_{p}^{p} − 1)^{1/2} .

Proof. Suppose first that M(f) = 1, that is, ν_{0}(f) = 0. Write ϕ = log |f|.

For every real t we have

e^{t} ≥ \{ \begin{matrix}1 + t + t^{2}/2 & if t ≥ 0, \\ 1 + t & if t < 0. \end{matrix}

This can be written as

e^{t} ≥ 1 + t + \frac{1}{2}t_{+}^{2}

for all t. In particular we have

|f|^{p} = e^{pϕ} ≥ 1 + pϕ + \frac{p^{2}}{2} ϕ_{+}^{2}.

Integrating this inequality and taking into account that ∫ ϕ = 0 we obtain

∥f∥_{p}^{p} ≥ 1 + \frac{p^{2}}{2} ∫ ϕ_{+}^{2},

or

∫ ϕ_{+}^{2} ≤ \frac{2}{p^{2}} (∥f∥_{p}^{p} − 1) .

By Schwarz’s inequality we obtain

∫ ϕ_{+} ≤ \frac{\sqrt{2}}{p} (∥f∥_{p}^{p} − 1)^{1/2} .

Since ∫ ϕ = 0, we have

∫ |ϕ| = 2 ∫ ϕ_{+} ≤ \frac{2\sqrt{2}}{p} (∥f∥_{p}^{p} − 1)^{1/2} .

This concludes the proof of the special case. The general case follows by applying the settled case for the function f/M(f).

Lemma 3.5.

For every p ∈ (0, ∞) and nonzero integer vector k we have

|ν_{k}(f)| ≤ \frac{\sqrt{8}}{p} (∥\frac{f}{M(f)}∥_{p}^{p} − 1)^{1/2} .

Proof. We have obviously

|ν_{k}(f)| ≤ Δ(f)

for every function f whenever k ≠ 0 and the assertion follows from the previous lemma.

Proof of Theorem 2, part a).

Suppose that there is a 0 < p ≤ ∞ such that for every ε there is a polynomial g such that g(0) = 1 and ∥fg∥_{p} < (1 + ε)M(f). By the monotonicity of ∥...∥_{p} we may assume that p < ∞.

Since

\frac{∥fg∥_{p}}{M(f)} = \frac{∥fg∥_{p}}{M(fg)}M(g)

and both factors are ≥ 1, both factors are ≤ 1 + ε.

Let k be a mixed vector. By applying Lemma 3.3 for the function g we obtain |ν_{k}(g)| ≤ ν_{0}(g) = log M(g) ≤ log(1 + ε) < ε.

By applying Lemma 3.5 for the function fg we obtain

|ν_{k}(fg)| ≤ \frac{\sqrt{8}}{p} (∥\frac{fg}{M(fg)}∥_{p}^{p} − 1)^{1/2} ≤ \frac{\sqrt{8}}{p}((1 + ε)^{p} − 1)^{1/2}.

Hence

|ν_{k}(f)| = |ν_{k}(fg) − ν_{k}(g)| ≤ ε + \frac{\sqrt{8}}{p}((1 + ε)^{p} − 1)^{1/2}.

Since this holds for every ε, we conclude that ν_{k}(f) = 0.

Corollary 3.6. The polynomials 1 + x + y, x + cy do not belong to the Szegő class.

Indeed, a routine calculation (though not a very short one for 1 + x + y) reveals that ν_{1,−1} ≠ 0 for these polynomials.

4. Roots and the Szegő class

The Szegő property is also connected to the location of zeros.

Theorem 3.

If a polynomial f can be written in the form f = gh, where g, h are polynomials such that g has no root in the domain |x_{1}| < 1, ..., |x_{n}| < 1 and h has no root in the domain |x_{1}| > 1, ..., |x_{n}| > 1, then it is in the Szegő class.

We start with some lemmas.

Lemma 4.1. Let f be a polynomial of degree m in one variable. Suppose that f has no root in

the domain |x| < 1. Then for 0 ≤ r ≤ 1 and |x| ≤ 1 we have

|f(rx)| ≥ (\frac{1 + r}{2} )^{m} |f(x)|.

Proof. Write f in the form

f(x) = c ∏(1 − α_{j}x)

with |α_{j}| ≤ 1. If f(x) ≠ 0, then this gives a product representation for f(rx)/f(x), and we get the result by applying Lemma 2.2 to each factor. If f(x) = 0, then the statement holds trivially.

Lemma 4.2. Let f be a polynomial of degree m_{1}, ..., m_{n} in x_{1}, ..., x_{n} and suppose that f has no

root in the region |x_{1}| < 1, ..., |x_{n}| < 1. Let 0 ≤ r_{1}, ..., r_{n} ≤ 1. We have

|f(r_{1}x_{1}, ..., r_{n}x_{n})| ≤ ∏ (\frac{1 + r_{j}}{2} )^{m_{j}} |f(x)|.

Proof. This follows from the previous lemma by an easy induction.

Lemma 4.3.

Any polynomial g which has no root in the domain |x_{1}| < 1, ..., |x_{n}| < 1 is in the Szegő class.

Proof. Assume that g(0) = 1. Take a number 0 < r < 1 and consider the function g_{r}(x) = g(rx). This function has no root in |x_{j}| < 1/r, thus 1/g_{r} is regular here and we can approximate it by a polynomial G_{0} in the compact set |x_{j}| ≤ 1; say, we can achieve that |g_{r}G_{0} − 1| < δ. This imples 1 − δ < |g_{r}G_{0}| < 1 + δ. If we put now G(x) = G_{0}(x)/G_{0}(0), we have G(0) = 1 and

|g_{r}G| < (1 + δ)/(1 − δ).

By virtue of Lemma 4.2 we have

|gG| < (\frac{1 + δ}{1 − δ} ) (\frac{2}{r + 1})^{m} ,

where m = m_{1} + ... + m_{n}. Since δ can be arbitrary small and r can be arbitrary near to 1, we conclude that S_{∞}(g) ≤ 1 = |g(0)| ≤ M(g) (the last inequality is contained in Lemma 2.1), thus g is in the Szegő class.

The general case follows by considering g(x)/g(0).

Lemma 4.4.

Let f be a polynomial of degree m in one variable. Suppose that f has no root in the domain |x| < 1. Then

|f(0)| ≥ 2^{−m} max_{|x|≤1} |f(x)|.

Proof. This follows from Lemma 4.1 by putting r = 0,

Lemma 4.5.

Let g be a polynomial in n variables. Suppose that g is not divisible by any of the polynomials x_{j}, and all roots in the domain |x_{1}| < 1, ..., |x_{n}| < 1 satisfy x_{1}x_{2}...x_{n} = 0. Then g has no root in this domain at all.

Proof. We use induction on n. For n = 1 the statement is obvious. Suppose we already know it for polynomials in n − 1 variables. We can write g in the form

g = G_{0} + x_{1}G_{1} + ... + x_{1}^{m}G_{m},

where the G_{j} are polynomials in x_{2}, ..., x_{n}.

Fix the value of x_{2}, ..., x_{n} so that x_{2}...x_{n} ≠ 0 and consider the polynomial
f(x) = G_{0} + G_{1}x + ... + G_{m}x^{m}.

By our assumption on g the only root of f in the disk |x| ≤ 1 can be x = 0. If this is not a root either, then Lemma 4.3 yields

|G_{0}(x_{2}, ..., x_{n})| ≥ 2^{−m} max_{|x_{1}|≤1} |g(x_{1}, ..., x_{n})|.  (4.1)

Thus (4.1) holds whenever G_{0} ≠ 0. But G_{0} is not identically zero, hence its roots are nowhere dense and consequently by continuity (4.1) holds everywhere.

By (4.1) we know that all roots of G_{0} in the domain |x_{2}| < 1, ..., |x_{n}| < 1 satisfy x_{2}...x_{n} = 0. By the inductive hypothesis we conclude that G_{0} has no roots at all. This means that g has no roots satisfying x_{1} = 0. The same argument, applied in turn to the variables x_{2}, ..., x_{n}, shows that g has no root in our domain.

Lemma 4.6.

Any polynomial h which has no root in the domain |x_{1}| > 1, ..., |x_{n}| > 1 is in the Szegő class.

Proof. Let m_{1}, ..., m_{n} be the degrees of h in x_{1}, ..., x_{n}, respectively. Then
g(x) = x_{1}^{m_{1}}...x_{n}^{m_{n}}h(x_{1}^{−1}, ..., x_{n}^{−1})

will be a polynomial, not divisible by any x_{j}. By definition it has no root satisfying 0 < |x_{1}|, ..., |x_{n}| < 1, thus by Lemma 4.5 it has no root satisfying |x_{j}| < 1. By Lemma 4.3 g is in the Szegő class. Thus we can find a polynomial G such that ∥gG∥_{∞} < (1+ε)M(g). Consider

H(x) = x_{1}^{q_{1}}...x_{n}^{q_{n}}G(x_{1}^{−1}, ..., x_{n}^{−1}),

where q_{j} is the degree of G in x_{j}. This defines a polynomial, and we have

h(x)H(x) = x_{1}^{q_{1}−m_{1}}...x_{n}^{q_{n}−m_{n}}g(x_{1}^{−1}, ..., x_{n}^{−1})G(x_{1}^{−1}, ..., x_{n}^{−1}).

Hence

∥hH∥_{∞} = ∥gG∥_{∞} < (1 + ε)M(g) = (1 + ε)M(h),

which shows that h is in the Szegő class.

Proof of Theorem 3. With f = gh, the polynomial g is in the Szegő class by Lemma 4.3 and h is there by Lemma 4.6. Theorem 2 shows that the Szegő class is closed under multiplication (or, directly, we can use the product GH of the polynomials constructed for g and h).

I conjecture that the sufficient condition provided by Theorem 3 for membership in the Szegő class is also necessary, that is, a polynomial f is in the Szegő class if and only if it can be written in the form f = gh, where g, h are polynomials such that g has no root in the domain |x_{1}| < 1, ..., |x_{n}| < 1 and h has no root in the domain |x_{1}| > 1, ..., |x_{n}| > 1.

This would also imply that the complement of the Szegő class is a multiplicative ideal.

I can establish the validity of the conjecture only for a very special case, which sounds as follows. Let q be a nonzero polynomial (in one variable y) without constant term and let f(x, y) = a + bx + q(y) with b ≠ 0. This polynomial f is in the Szegő class if and only if |b| < |a| and the polynomial q satisfies |q(y)| ≤ |a| − |b| for all |y| ≤ 1.

In particular, a linear polynomial a + bx + cy with bc ≠ 0 is in the Szegő class if and only if |b| + |c| ≤ |a|.

I suppress the proof of these results with the hope that someone will be able to decide a more general case.

Problem. The proof of Theorem 2 yields a lower estimate for S_{p}(f). It would be interesting to describe the exact value of S_{p}(f) (at least for p = 2). For p = 2 the Parseval formula expresses this as the infimum of a certain quadratic form (in infinitely many variables) but it seems difficult to gain any further information in this way.

References

[1] Smyth, C. J., A Kronecker-type theorem for complex polynomials in several variables, Canad. Math. Bull. 24(1981), 447–451; addendum and errata ibid. 25 (1982), 504.

[2] Szegő, G., Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion, Math. Ann. 76(1915), 490–503. Printed 1999.2.28.