LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS

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1 LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS MEHDI GHASEMI AND MURRAY MARSHALL Abstract. Recently Lasserre [6] gave a sufficient condition in terms of the coefficients for a polynomial f R[X] of degree (d 1) in n 1 variables to be a sum of squares of polynomials. Exploiting this result, we are able to determine, for any polynomial f R[X] of degree whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f r is a sum of squares of polynomials. The existence of such a number r was proved earlier by Marshall [8], but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite. Similar arguments are applied to results of Fidalgo and Kovacec [2], to determine other numbers r such that f r is a sum of squares and other lower bounds for f. 1. Introduction Fix a non-constant polynomial f R[X] = R[X 1,, X n ], where n 1 is an integer number, and define f = inf{f(a) a R n }. Let deg(f) = m and decompose f as f = f f m (the homogeneous decomposition of f), where f i is a form of degree i, i = 0,..., m. A necessary condition for f is that f m is positive semidefinite (so, in particular, m is even). A sufficient condition for f is that f m is positive definite. We assume from now on that deg(f) =, d 1, i.e., m =. For convenience we denote the cone of all sos 1 polynomials by R[X] 2 and the cone of all positive semidefinite forms and sos forms of degree by P,n and Σ,n. Also, by R[X] k we mean the subspace of R[X] consisting of all polynomials with degree at most k. Define (1) f sos = sup{r R f r R[X] 2 }. One can prove that f sos f and (2) f sos = inf{l(f) l χ }, where χ is the set of all linear maps l : R[X] R such that l(1) = 1 and l(p 2 ) 0 for all p R[X] of degree d. Computing f is difficult in general, and one of the successful approaches is to compute f sos instead. This is accomplished by using semidefinite programming (SDP) which is a polynomial time algorithm [5] [9]. The equivalent definitions (1) and (2) for f sos can be considered as two SDP problems which are dual of each other (and the duality gap is 0 in this case) Mathematics Subject Classification. Primary 12D99 Secondary 14P99, 90C22. Key words and phrases. Positive polynomials, sums of squares, optimization. 1 Abbreviation for sum of squares 1

2 2 MEHDI GHASEMI AND MURRAY MARSHALL When is a given polynomial f R[X] a sum of squares? One obvious necessary condition is that f 0 on R n, but there is a well known result due to Hilbert [3] that, in general, for n 3, d 2 and (n, d) (3, 2), P,n Σ,n. The Motzkin polynomial s(x, Y ) = 1 3X 2 Y 2 + X 4 Y 2 + X 2 Y 4 is a concrete example of a polynomial which is non-negative on R 2 but is not a sum of squares. In this paper we will be interested in some recent results, due to Lasserre [6] and to Fidalgo and Kovacec [2], which give sufficient conditions on the coefficients for a polynomial to be a sum of squares. We denote by P,n and Σ,n the interior of P,n and Σ,n, more precisely, the interior in the subspace of R[X] consisting of forms of degree. A necessary condition for f sos is that f Σ,n. A sufficient condition for f sos is that f Σ,n [8, Prop. 5.1]. Scaling X by a non-zero scalar k does not change the value of f sos. In Sections 2 and 3 we show that if f = n X i there are suitable choices for k and r such that f(kx) r satisfies Lasserre s condition in [6, Th. 3] and (after homogenization) Fidalgo-Kovacec s condition in [2, Th. 4.3]. This allows us to determine, assuming that f Σ,n, two lower bounds for f sos, which we denote by r L and r F K respectively; see Ths. 2.3 and 3.1. Yet another lower bound for f sos, which we denote by r dmt, is obtained by applying [2, Th. 2.3] more or less directly; see Th The bounds r L, r F K and r dmt are not comparable; see Ex If we assume only that f P,n then it is still possible to determine lower bounds r L, r F K and r dmt for f, but these may not be lower bounds f sos ; see Th We introduce more notation that we will need. Let N = {0, 1, 2,...} be the set of natural numbers. For X = (X 1,..., X n ) and α = (α 1,..., α n ) N n, define X α := X α1 1 X α n n and α := α 1 + +α n. Clearly, using this notation, every polynomial f R[X] can be written as f(x) = α N f αx α, where f n α R and f α = 0, except for finitely many α. Let Ω(f) = {α N n f α 0} \ {0, ɛ 1,..., ɛ n }, where = deg(f), ɛ i = (δ i1,..., δ in ), and { 1 i = j δ ij =. 0 i j We denote f 0 and f ɛi by f 0 and f,i for short. Thus f has the form (3) f = f 0 + n f α X α + f,i Xi. α Ω(f) Let (f) = {α Ω(f) f α X α is not a square in R[X]} = {α Ω(f) either f α < 0 or α i is odd for some i {1,..., n}}. Since our polynomial f is usually fixed, we will often denote Ω(f) and (f) just by Ω and for short. Let f(x, Y ) = Y f( X 1 Y,..., X n Y ). From (3), it is clear that f(x, Y ) = f 0 Y + α Ω f α X α Y α + is a form of degree, called the homogenization of f. well-known result: Proposition 1.1. f is sos if and only if f is sos. Proof. See [7, Prop ]. n f,i X i We have the following

3 LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS 3 For a (univariate) polynomial of the form p(t) = t n n 1 i=0 a it i, where each a i is nonnegative and at least one a i is nonzero, we denote by C(p) the unique positive root of p [10, Th ]. For any polynomial q(t) = n i=0 b it i, b n 0, the roots of q are bounded in absolute value by C(t n n 1 i=0 b i b n ti ). By convention, C(t n ) := 0. There are various upper bounds for C(p) which are expressible in an elementary way in terms of the coefficients of p, for example, Proposition 1.2. Suppose p(t) = t n n 1 i=0 a it i, where each a i is nonnegative and at least one a i is nonzero. Then (1) C(p) max{1, a 0 + a a n 1 }, (2) C(p) max{a 0, 1 + a 1, 1 + a 2,..., 1 + a n 1 }, (3) C(p) 2 max{a n 1, (a n 2 ) 1/2, (a n 3 ) 1/3,..., (a 0 ) 1/n }. Proof. Bounds (1) and (2) are due basically to Cauchy. See [1] for these bounds and for other bounds of this sort. See [4, Ex : 20] for bound (3). 2. Application of Lasserre s Result Lasserre proved in [6] that for a given f R[X] of degree, if (L1) f 0 f α and (L2) α min f,i f α α,...,n α then f R[X] 2 ; see [6, Th. 3]. Theorem 2.1. X X n Σ,n. Th. 2.1 is proved already in [8, Prop. 5.3(2)]. We give another proof here which is based on Lasserre s result. Th. 2.1 can also be deduced from Fidalgo-Kovacec s result [2, Th. 4.3]. See Section 3 below for the statement of [2, Th. 4.3]. Proof. Let p(x) = X1 + + Xn + h(x) where h(x) is any form of degree whose coefficients have absolute value ɛ where ɛ is some small positive real. One sees that, for ɛ sufficiently small, the polynomial q R[X 1,..., X n 1 ] defined by q(x 1,..., X n 1 ) = p(x 1,..., X n 1, 1) satisfies (L1) and (L2) so, by Lasserre s result, q is sos. Then, by Prop. 1.1, the homogenization of q (which is p) belongs to Σ,n. Remark 2.2. For any cone C in a finite dimensional real vector space V, if f C and g V then g C iff g ɛf C for some real ɛ > 0. Proof. Suppose g ɛf C. Let h V. Since f belongs to the interior of C, there exists some real δ > 0 such that f + δ ɛ h C. Then g+δh = (g ɛf)+ɛ(f + δ ɛ h) C. This proves that g belongs to the interior of C. The other implication is clear. Every polynomial f of degree decomposes as f = f 0 + f f (the homogeneous decomposition of f) where each f i is a form of degree i. The following theorem gives a sufficient condition for f sos in terms of the highest degree form, f, and a concrete lower bound for f sos :

4 4 MEHDI GHASEMI AND MURRAY MARSHALL Theorem 2.3. If f Σ,n then f sos r L, where r L := f 0 α, α < f α ɛ α k α, k := C(t and ɛ > 0 is given such that f ɛ( n X i ) Σ,n. α, α < f α α α ɛ t α ) Note: (i) If {α α < } =, then k = C(t ) := 0 and r L := f 0. (ii) Th. 2.3 proves in particular that if f Σ,n then f sos, i.e., it provides another proof of [8, Prop. 5.1]. (iii) If l k then f 0 α, α < f α ɛ α l α f 0 α, α < f α ɛ α k α = r L. In this way, by taking l to be one of the upper bounds for k mentioned in Prop. 1.2, we obtain a lower bound f 0 α, α < f α ɛ α l α for f sos which is expressible in an elementary way in terms of ɛ and the coefficients f α, α <. (iv) We are assuming here that ɛ is given. Refer to Th. 4.2 below for a method for computing ɛ which applies in certain cases. Proof. Since f Σ,n, by Th. 2.1 and Rem. 2.2, there exists ɛ > 0 such that f = ɛ(x1 + + Xn ) + g for some g Σ,n. Scaling suitably (X i X i ), we ɛ can assume that ɛ = 1. Let ˆf := f g. Decomposing ˆf as in equation (3) yields (4) ˆf = f0 + α Ω, α < f α X α + n X i. If {α α < } =, then ˆf r L = ˆf f 0 is sos, using equation (4) and the definition of, so f r L is also sos and the result is clear. Thus we can assume {α α < } =, so k > 0. Scaling by X i kx i, and rewriting condition (L2) for the polynomial ˆf(kX) r, using equation (4), yields k α, α < f α α k α. By definition of k, k = α, α < f α α k α, so condition (L2) holds for ˆf(kX) r. Rewriting (L1) for the polynomial ˆf(kX) r, we see that if r rl then (L1) holds for ˆf(kX) r so ˆf r is sos and hence also f r is sos. 3. Application of Fidalgo-Kovacec s Results Fidalgo and Kovacec in [2] proved that if f R[X] is a form of degree and (FK) min f,i 1,...,n n ( n ) f α α α α then f is a sum of squares [2, Th. 4.3]. Here, α α := α α1 1 ααn n (the convention being that 0 0 := 1). If f R[X] is arbitrary of degree then, rewriting (FK) for f (the homogenization of f), we see that if (FK ) min {f,i, f 0 } 1,...,n n+1 ( n+1 ) α f α α α ( α ) α

5 LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS 5 then f is sos, by Prop So f r is sos as long as min,,n {f,i, f 0 r} the right side of (FK ). One can use more or less the same idea used in the proof of Th. 2.3 to find a lower bound for f sos using (FK ) as follows: Theorem 3.1. If f Σ,n then f sos r F K, where r F K := f 0 k, k := C(t 1 b it i ), b i := 1 ( ) n + 1 ( i) i ɛ i f α α α, i = 1,..., 1 n + 1 and ɛ > 0 is given as in Th α, α =i Note: If l k then 1 1 f 0 b i l i f 0 b i k i = f 0 k = r F K so, using Prop. 1.2 again, we get another lower bound for f expressible in an elementary way in terms of ɛ and the coefficients f α, α <. Proof. After scaling we can assume that ɛ = 1 and f = X X n + g, where g Σ,n. If {α α < } =, then b i = 0 for i = 1,... 1, k = 0 (by definition of C(t )), so r F K = f 0. In this case the result is clear. So we can assume {α α < } =, so k > 0. Set r = r F K. Rewriting condition (FK ) for ˆf(kX) r, where ˆf := f g, yields the condition: (5) min{(f 0 r), k } 1 b i k i. By definition of k and r, (5) holds, in fact, f 0 r = k = 1 b ik i. This proves that ˆf r is sos and hence also that f r is sos. According to [2, Th. 2.3], a homogeneous polynomial n β ixi µx α, such that α i > 0 and β i 0 for every i = 1,..., n and µ 0 if all α i are even, is sos if and only if n ( ) α i βi (6) µ. Theorem 3.2. If f Σ,n then f sos r dmt := f 0 α, α < α i ( α ) [ (fα where t := {α α < } and ɛ > 0 is given as in Th ) ] 1 (( t α ɛ ) α α α ), Proof. Let = {α α < }. After scaling, we can assume that ɛ = 1. Let f = f 0 + α f αx α +X 1 + +Xn and let F (X, Y ) denote the homogenization of f( tx) r, where r := f 0 α r α, each r α 0. Then F (X, Y ) = (f 0 r)y Xn + f α t α / X α Y α ) α (X = (r α Y + X1 + + Xn + f α t α / X α Y α ). α

6 6 MEHDI GHASEMI AND MURRAY MARSHALL By (6), each term appearing in this sum will be sos if or, equivalently, if ( ) α f α t α rα α r α ( α ) [ (fα α i 0 ( 1 α i ) t α α α ] 1 α. ) α i, Hence if r r dmt then f r is sos, so also f r is sos. 4. Further remarks (1) The lower bounds r L, r F K and r dmt for f sos described in Ths. 2.3, 3.1 and 3.2 are not comparable. Example 4.1. (a) For f(x, Y ) = X 6 + Y 6 + 2X 3 Y 2 + 1, we have r L = 1 2( 5 3 ) , r F K = 1 ( 9 8 ) and r dmt = , so r dmt is best. (b) For f(x) = X 6 X 4 X 2, r L = 2, r F K and r dmt , so r L is best. (c) For f(x, Y, Z) = X 4 + Y 4 + Z 4 + axy Z + bxy, a, b > 0, r F K > r dmt > r L, so r F K is best, provided b is sufficiently small compared to a. (2) To be able to compute r L, r F K and r dmt one needs to know ɛ and the coefficients f α, α <. What can one do if ɛ is not given, i.e., if only the coefficients f α, α are given? The results in [2, Th. 4.3] and [6, Th. 3] provide sufficient conditions for f Σ,n to hold and have the nice additional property of allowing computation of ɛ: Theorem 4.2. Suppose ɛ := max{ɛ 1, ɛ 2 } > 0 where ɛ 1 := min f,i f α, ɛ 2 := min f,i 1,...,n,...,n n ( n ) α, α = Then f Σ,n and f ɛ n X i Σ,n. α, α = f α α α. Proof. Suppose ɛ = ɛ 2. The inequality (FK) holds for the form f ɛ n X i, by definition of ɛ 2, so f ɛ n X i is sos and f Σ,n by Rem Suppose ɛ = ɛ 1. Consider the dehomogenization h(x 1,..., X n 1 ) = f (X 1,..., X n 1, 1) ɛ( n 1 X i + 1) of f ɛ n X i. The inequality (L1) holds for h by definition of ɛ 1. The inequality (L2) holds using min (f,i ɛ 1 ),...,n 1 α, α = f α α, α = f α α α n. Thus h is sos. Rehomogenizing and applying Prop. 1.1 this implies that f ɛ n X i is sos. In [6, Cor. 5] it is proved that if ɛ 1 > 0 then f sos. Using Th. 4.2 we see that this can also be deduced from Th We remark also that, in the statement

7 LOWER BOUNDS FOR A POLYNOMIAL IN TERMS OF ITS COEFFICIENTS 7 of Th. 4.2, it is possible to improve slightly on the definition of ɛ 1. Namely, one can take ɛ 1 := max{ɛ(1),..., ɛ(n)} where ɛ(i) := min{f,i α, α = f α, min j i f,j α, α = f α α α i }. This is more or less clear from the proof of Th If ɛ = ɛ 1 = ɛ(i) just dehomogenize f ɛ n j=1 X j with respect to the variable X i and apply Lasserre s [6, Th. 3] to deduce that f ɛ n j=1 X j is sos. (3) So far we have been assuming that f Σ,n and we have used this assumption to determine lower bounds for f sos. What can one say if one assumes only that f P,n? Suppose ɛ > 0 is given such that f ɛ n X i P,n. One can then define r L exactly as in Th. 2.3, but using this new ɛ, i.e., r L := f 0 f α ɛ α k α, k := C(t f α α t α ). α, α < α, α < α ɛ The r L defined in this way might not be a lower bound for f sos (it is even possible that f sos = ), but it will be a lower bound for f. Similar remarks apply to the other bounds r F K and r dmt. Theorem 4.3. If f P,n and ɛ > 0 is such that f ɛ n X i P,n then r L, r F K and r dmt, defined as in Ths. 2.3, 3.1 and 3.2, respectively, but using this new choice of ɛ, are lower bounds for f on R n. Proof. Argue as in the proof of Ths. 2.3, 3.1 and 3.2. The form g is no longer sos but it is positive semidefinite, which is all one needs for the conclusion. Note: In Th. 4.3, the largest possible choice for ɛ is the minimum value of the rational function f / n X i on the n 1-sphere S n 1 := {a R n a a 2 n = 1}. (4) We know that for any p P,n and any g R[X] 1, (p + g) and, for any p Σ,n and any g R[X] 1, (p + g) sos. Note that if p P,n is not positive definite then there exists 0 a R n such that p(a) = 0. Let g(x) = n a ix i. Then (p + g)(ta) = t a 2 as t, so (p + g) =. Therefore for any p P,n ( P,n denotes the boundary of P,n, i.e. P,n = P,n \ P,n ), there exists g R[X] 1, such that (p + g) =. The validity of the corresponding result for boundary points of Σ,n is unknown to the authors. Question 4.4. Is it true that for any p Σ,n there exists g R[X] 1 such that (p + g) sos =? The answer to this question is yes if n 2 or d = 1 or (n = 3 and d = 2) by Hilbert s result [3]. In fact these are precisely the cases where P,n and Σ,n coincide. Acknowledgement The first author expresses his gratitude to Prof. S. Kuhlmann for introducing him to this general area of research and encouraging him to work on it.

8 8 MEHDI GHASEMI AND MURRAY MARSHALL References [1] E. Deutsch, Bounds for the zeros of polynomials, Amer. Math. Monthly 88 ( ), [2] C. Fidalgo, and A. Kovacec, Positive semidefinite diagonal minus tail forms are sums of squares, Preprint, [3] D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann. 32 ( ), [4] D. Knuth, The Art of Computer Programming, Volume 2, Addison-Wesley, New York, [5] J. B. Lasserre, Global Optimization with Polynomials and the Problem of Moments, SIAM J. Optim. Volume 11, Issue 3 ( ), [6] J. B. Lasserre, Sufficient Conditions for a Real Polynomial to be a Sum of Squares, Arch. Math. (Basel) 89 ( ), [7] M. Marshall, Positive Polynomials and Sum of Squares, Mathematical Surveys and Monographs, Vol.146, [8] M. Marshall, Representation of Non-Negative Polynomials, Degree Bounds and Applications to Optimization, Canad. J. Math., 61 ( ), [9] P. A. Parrilo, B. Sturmfels, Minimizing Polynomial Functions, Ser. Discrete Math. Theor. Comput. Sci. 60 (83-99), [10] V. V. Prasolov, Polynomials, Algorithms and Computation in Mathematics Vol.11, Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6, Canada address: mehdi.ghasemi@usask.ca, marshall@math.usask.ca