# Homogeneous Polynomial Systems

## Abstract

We finished the preceding chapter with a notion of approximate zero of a function and an algorithmic scheme to compute these approximate zeros, the adaptive homotopy.

Within this scheme, we identified as critical the issue of determining the step length at each iteration of the continuation process, and as a first step towards this goal, we estimated the size of the basin of quadratic attraction of a proper zero *ζ* for a given analytic function *f*: this basin contains a ball of radius \(\frac{3-\sqrt{7}}{2\gamma(f,\zeta )}\) centered at *ζ*.

At this stage we perceive two weaknesses in this estimate. Firstly, the computation of *γ*(*f*,*ζ*) appears to require the computation of the norm of all the higher order derivatives of *f* at *ζ*. Even if we deal with polynomials (for which the number of such computations is finite), this can be very costly. Secondly, we can hardly compute these derivatives without having *ζ* at our disposal. And the whole idea of the adaptive homotopy relies on not having resource to the zeros *ζ* _{ t } in the lifted path *Γ*.

In this chapter we provide solutions for these shortcomings. To do so, we narrow the context we are working on and focus on a specific class of functions, namely homogeneous multivariate polynomial functions \(f: \mathbb {C}^{n+1}\to \mathbb {C}^{n}\). Homogenization is a common approach to the study of zeros for not necessarily homogeneous polynomial systems: given one such system, one homogenizes its component polynomials and considers the zeros of the resulting homogeneous system, which are now sets of lines through the origin, as points in projective space \(\mathbb {P}^{n}\). In doing so, one avoids the distortions produced by having “large” zeros or, in the limit, zeros at infinity. We denote by \(\mathcal {H}_{\mathbf{d}}\) the linear space of homogeneous polynomial systems with degree pattern **d**=(*d* _{1},…,*d* _{ n }).

Newton’s method as described in the previous chapter can be modified to work in this setting (i.e., acting on \(\mathbb {P}^{n}\) and with underlying function in \(\mathcal {H}_{\mathbf{d}}\)). With a few natural modifications we recover both the notion of approximate zero and a version *γ* _{proj}(*f*,*ζ*) of the *γ* invariant. Furthermore—and gratifyingly, also with only a few minor modifications—we show that the size of the basin of quadratic attraction of a zero is controlled by *γ* _{proj} in about the same manner as what we saw in Chap. 15.

The invariant *γ* _{proj}(*f*,*ζ*) is also defined in terms of higher-order derivatives and therefore shares the first weakness mentioned above. Condition proves helpful to overcoming it. The solution in \(\mathbb {P}^{n}\) of systems in \(\mathcal {H}_{\mathbf{d}}\) fits within the framework described in Chap. 14 and therefore, to a pair \((f,\zeta )\in \mathcal {H}_{\mathbf{d}}\times \mathbb {P}^{n}\) with *f*(*ζ*)=0 we may associate a condition number *μ*(*f*,*ζ*). Shub and Smale introduced a normalization of *μ*(*f*,*ζ*)—denoted by *μ* _{norm}(*f*,*ζ*)—whose value is close to *μ*(*f*,*ζ*) and that allows for some elegant statements, such as a condition number theorem.

We see that *γ* _{proj}(*f*,*ζ*) can be bounded in terms of *μ* _{norm}(*f*,*ζ*). To solve the second shortcoming, a key step is the observation that the condition *μ* _{norm} satisfies a *Lipschitz property* that allows one to estimate *μ* _{norm}(*g*,*y*) in terms of *μ* _{norm}(*f*,*z*) for pairs (*g*,*y*) close to (*f*,*z*).