MATHEMATICS OF COMPUTATION Volume 70, Number 233, Pages 329{335 S 0025-5718(00)00934-0 Article electronically published on July 10, 2000 A CONDITION NUMBER THEOREM FOR UNDERDETERMINED POLYNOMIAL SYSTEMS JER´ OME^ DEGOT´ Abstract. The condition number of a numerical problem measures the sen- sitivity of the answer to small changes in the input. In their study of the complexity of B´ezout's theorem, M. Shub and S. Smale prove that the condi- tion number of a polynomial system is equal to the inverse of the distance from this polynomial system to the nearest ill-conditioned one. Here we explain how this result can be extended to underdetermined systems of polynomials (that is with less equations than unknowns). 1. Introduction The condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. For a given problem, a condition number theorem asserts that the condition number µ is equal to the inverse of the distance of that problem to the set Σ of ill-posed ones. A first example of such a theorem is due to Eckart and Young [4] about the problem of matrix inversion. See Demmel [3] and Dedieu [2] for a general study concerning condition number theorems for various numerical problems like: matrix inversion, computing eigenvalues and eigenvectors, finding zeroes of polynomials, pole assignment in linear control, ::: In the case of polynomial systems with the same number of equations as un- knowns, Shub and Smale [6] have proved a condition number theorem. We will give a direct and elementary proof of this theorem, using various properties of Bombieri's scalar product. This allows us to extend their result to the case of underdetermined polynomial systems (that is with less equations than unknowns). 2. Background Here, we deal only with homogeneous polynomials. Extensions to the affine case are generally trivial, fixing the first variable to 1. Let Hd denote the linear space of all homogeneous polynomial systems P = n+1 m (P1;:::;Pm): C ! C ,whereeachPi is a homogeneous polynomial of n +1- variables, of degree di and d =(d1;:::;dm). Remark 1. We denote by n + 1 the number of variable, because the first variable n m z0 is added to homogenize an ordinary affine polynomial system C ! C . Received by the editor August 13, 1996. 2000 Mathematics Subject Classification. Primary 65H10. c 2000 American Mathematical Society 329 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 330 JER´ OME^ DEGOT´ n+1 For two homogeneous polynomials Pi;Qi : C ! C of degree di, Bombieri's scalar product is defined by X α! [Pi;Qi](di) = aαbα; di! jαj=d P P i where P (z)= a zα, Q (z)= b zα. With the usual notations, i jαj=di α i jαj=di α n+1 if α =(α0,...,αn) 2 N ,wehavejαj = α0 + ··· + αn, α!=α0! :::αn!and α α0 αn H 2H z = z0 :::zn . This induces a hermitian inner product on d for P; Q d, Xm [P; Q]= [Pi;Qi](di); i=1 we denote by kP k and d(P; Q) the associated norm and distance defined by 1 kP k =[P; P] 2 and d(P; Q)=kP − Qk : We now give a series of lemmas concerning Bombieri's scalar product. If x 2 Cn+1, we use δx to denote the homogeneous (n + 1)-variable polynomials of degree 1, defined by Xn δx(z)= xizi : i=0 Remark 2. If x; y 2 Cn+1,wehave h i [δx,δy](1) = y;x ; where h:; :i is the usual inner product of Cn+1. The following lemma allows us to replace the evaluation of a polynomial at a point by a duality formula. Lemma 1 (B. Reznick [5]). Let P : Cn+1 ! C be a homogeneous polynomial of degree d. For every x 2 Cn+1 we have d P (x)=[P; δx](d): Lemma 2 (B. Reznick [5]). Let P,Q and R be three homogeneous polynomials of respective degrees p, q and r such that p + q = r; then q! [PQ;R] = [Q; P (D)R] ; (r) r! (q) where P(D) is the differential operator defined by @ @ P (D)=P ( ;:::; ): @x0 @xn The next lemma is a corollary of Proposition 4 of Beauzamy-D´egot [1]. Lemma 3. Let P1;:::;Pk and Q1;:::;Qk be homogeneous polynomials of degree 1. We have 1 X [P :::P ;Q :::Q ] = [P ;Q ] ×···×[P ;Q ]; 1 k 1 k (k) k! 1 σ(1) k σ(k) σ where σ runs over the set Sk of all the permutations of f1;:::;kg. Combining Lemmas 1 and 2, we obtain a duality formula for polynomial systems. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A CONDITION NUMBER THEOREM 331 n+1 m Lemma 4. Let P =(P1;:::;Pm):C ! C be a homogeneous polynomial n+1 system, such that each Pi is of degree di.Forx; y 2 C we have − DP(x)yj = d [P ,δdi 1δ ] for all i =1;:::;m; i i i x y (di) where DP(x)yji denotes the ith coordinate of DP(x)y, the differential of P at x applied to y. Proof. For all i =1;:::;m,wehave Xn @P DP(x)yj = y i (x): i j @x j=0 j Using Lemmas 1 and 2, we obtain − j di 1 DP(x)y i =[δy(D)Pi,δx ](di−1) di−1 = di [Pi,δx δy](di) ; which proves the lemma. For the proof of the first three lemmas and for a detailed study of the properties of Bombieri's scalar product, we refer the reader to [1]. 3. The condition number of a polynomial system We consider the numerical solution of the equation P (x)=0 where P : Cn+1 ! Cm is a homogeneous polynomial system. We want to define the condition number of the system P . The condition number should measure the relative sensitivity of the solution (output) with respect to the change of the data (input). Let ∆P be an infinitesimal perturbation of P ,andlet∆x be the first order corresponding perturbation of x, in the following sense: (1) k∆xk'inffky − xk ;(P +∆P )(y)=0g: ? Lemma 5. The previous formula implies that ∆x 2 (KerDP(x)) . Lemma 6. We have ∆x = −DP(x)∗∆P (x) ; where DP(x)∗ is the Moore-Penrose inverse of DP(x), that is ∗ j −1 DP(x) =(DP(x) (KerDP(x))? ) : Proofs. The Taylor expansion of P +∆P gives 0=(P +∆P )(x +∆x) = P (x +∆x)+∆P (x +∆x) = P (x)+DP(x)∆x +∆p(x)+o(k∆xk) : ? Then DP(x)∆x = −∆P (x). By relation (1), we deduce that ∆x 2 (KerDP(x)) , andsowehave∆x = −DP(x)∗∆P (x), which gives the lemma. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 332 JER´ OME^ DEGOT´ We can now compute the condition number: k∆xk kDP(x)∗∆P (x)k = kxk kxk − kDP(x)∗Diag(kxkdi )Diag(kxk di )∆P (x)k = kxk where Diag(kxkdi ) is the diagonal matrix such that Diag(kxkdi )=Diag(kxkd1 ;:::;kxkdm ): Then ∗ di k∆xk kDP(x) Diag(kxk )k − ≤ op kDiag(kxk di )∆P (x)k : kxk kxk 2 By Lemma 1, we have j j j di |≤ k kdi Pi(x) = [Pi,δx ] [Pi] x ; so − k k k di k ≤k k Diag( x )∆P (x) 2 ∆P : Therefore, k∆xk − ≤kDP(x)∗Diag(kxkdi 1)k k∆P k : kxk op The condition number of P at x is then given by − k ∗ k kdi 1 k µ(P; x)= DP(x) Diag( x ) op : pFor sharper estimates on complexity it is convenient to have a further factor of di. Thus like in Shub-Smale [6], we will use the following normalization of the condition number p − k ∗ k kdi 1 k µnorm(P; x)= DP(x) Diag( di x ) op : 4. Condition number theorem It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of the problem and the shortest distance from that problem to an ill-posed one. A condition number theorem is a theorem which characterizes this relationship. We will give here a general condition number theorem for the problem of finding a zero of a polynomial system of equations. Definition 1. Let d =(d1;:::;dm) be fixed. We denote by Σx the set of all ill-posed problems at x,thatis Σx = fQ 2Hd ; Q(x)=0andtherankofDQ(x)islessthanmg: Theorem 1. Let P : Cn+1 ! Cm be a homogeneous polynomial system with n ≥ m,andletx 2 Cn+1 be such that P (x)=0and Rank(DP(x)) = m.Then 1 µnorm(P; x)= : d(P; Σx) Remark 3. In the case where n = m, this is the condition number theorem of Shub-Smale [6]. We will give here a proof of this theorem, using the properties of the linear space Hd equipped with the hermitian structure described in section 2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A CONDITION NUMBER THEOREM 333 5. Proof of the theorem We have d(P; Σx)= inf kP − Rk =infkQk : R2Σx (P +Q)(x)=0 Rank(D(P +Q)(x))<m Then we want to find the polynomial system Q 2Hd of smallest norm, such that (2) (P + Q)(x)=0; (3) Rank(D(P + Q)(x)) <m: 1 5.1.