Bounds on Distance to Variety in Terms of Coefficients of Bivariate Polynomials

TL;DR

This paper discusses bounds on the distance from a point to a variety, expressed in terms of the Taylor coefficients of bivariate polynomials at that point.

Contribution

It introduces bounds relating the distance to a variety with the Taylor coefficients of bivariate polynomials, providing new insights into polynomial geometry.

Findings

Derived bounds on distance to variety using Taylor coefficients

Applicable to bivariate polynomial cases

Offers a theoretical framework for polynomial approximation

Abstract

A short note on bounds on distance to variety of a point in terms of the Taylor coefficients at the point.

Equations

$$\frac{f^{\prime }(z)}{f(z)}=\sum _{i=1}^d\frac{1}{|z-{\alpha }_i|}.$$

$$\frac{f^{(k)}(z)}{f(z)}=\sum _{1\le i_1<i_2<\cdots <i_k\le d}\frac{1}{(z-{\alpha }_{i_1})(z-{\alpha }_{i_2})\dots (z-{\alpha }_{i_k})}.$$

$${\left|\frac{f^{(k)}(z)}{f(z)}\right|}^{1/k}\le \frac{d}{\mathrm{sep}(f,z)}.$$

$$\mathrm{sep}(f,z)\le d\underset{1\le k\le d}{\min }\left|\frac{f(z)}{f^{(k)}(z)}\right|.$$

$$f_{i,k}({\bf p})\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;\frac{\partial^{k}f({\bf p})}{\partial^{i}x\partial^{k-i}y}.$$

$$\mathrm{sep}({\bf p},V(f))\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;\inf_{{\bf x}\in V(f)}\|{\bf p}-{\bf x}\|$$

$$\mathrm{sep}(\mathbf{p},V(f))\le D\underset{1\le k\le D}{\min }{\left|\frac{f(\mathbf{p})}{f_{0,k}(\mathbf{p})}\right|}^{1/k}.$$

$$\mathrm{sep}(\mathbf{p},V(f))\le \underset{1\le k\le D}{\min }{\left|k!\left(\genfrac{}{}{0ex}{}{D}{k}\right)\frac{f(\mathbf{p})}{f_{k,0}(\mathbf{p})}\right|}^{1/k}.$$

$$f(x,y)\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;K\prod_{i=1}^{d(y)}(x-\alpha_{i}(y)),$$

$$f_{1,0}(x,y)=f(x,y)\sum _{i=1}^{d(y)}\frac{1}{x-{\alpha }_i(y)},$$

$$f_{k,0}(x,y)=f(x,y)\sum _{1\le i_1<i_2<\cdots <i_k\le d(y)}\frac{1}{(x-{\alpha }_{i_1}(y))\dots (x-{\alpha }_{i_k}(y))}.$$

$${\left|\frac{f_{k,0}(\mathbf{p})}{f(\mathbf{p})}\right|}^{1/k}\le \frac{D}{\mathrm{sep}(\mathbf{p},V(f))},$$

$$\left[\begin{array}[]{ccccccccccccccccccccccccc}x\\ y\\ \end{array}\right]\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccccccccccccccccccccccccc}e^{\jmath\theta}&e^{-\jmath\psi}\\ e^{-\jmath\psi}&-e^{-\jmath\theta}\\ \end{array}\right]\left[\begin{array}[]{ccccccccccccccccccccccccc}X\\ Y\\ \end{array}\right]=U\cdot\left[\begin{array}[]{ccccccccccccccccccccccccc}X\\ Y\\ \end{array}\right]$$

$$UU^{\dagger}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccccccccccccccccccccccccc}e^{\jmath\theta}&e^{-\jmath\psi}\\ e^{\jmath\psi}&-e^{-\jmath\theta}\\ \end{array}\right]\cdot\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccccccccccccccccccccccccc}e^{-\jmath\theta}&e^{\jmath\psi}\\ e^{-\jmath\psi}&-e^{\jmath\theta}\\ \end{array}\right]=\left[\begin{array}[]{ccccccccccccccccccccccccc}1&0\\ 0&1\\ \end{array}\right].$$

$$F(X,Y)\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;f(U(X,Y))=f\left({\frac{e^{\jmath\theta}X+e^{-\jmath\psi}Y}{\sqrt{2}},\frac{e^{\jmath\psi}X-e^{-\jmath\theta}Y}{\sqrt{2}}}\right).$$

$$F_{k,0}(X,Y)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{k}{i}\right)(f_{i,k-i}\circ U(X,Y)){\left(\frac{\partial x}{\partial X}\right)}^i{\left(\frac{\partial y}{\partial X}\right)}^{k-i}.$$

$$\partial x/\partial X=e^{ȷ\theta }/\sqrt{2},\partial y/\partial X=e^{ȷ\psi }/\sqrt{2}.$$

$$F_{k,0}(X,Y)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{k}{i}\right)(f_{i,k-i}\circ U(X,Y))e^{ȷi\theta }{3}^{ȷ(k-i)\psi }{2}^{-k}.$$

$${\left|\frac{F_{k,0}(U^{-1}(\mathbf{p}))}{F(U^{-1}(\mathbf{p}))}\right|}^{1/k}\le \frac{D}{\mathrm{sep}(U^{-1}(\mathbf{p}),V(F))}=\frac{D}{\mathrm{sep}(U^{-1}(\mathbf{p}),U^{-1}(V(f)))}=\frac{D}{\mathrm{sep}(\mathbf{p},V(f))},$$

$${\left|\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{k}{i}\right)\frac{f_{i,k-i}(\mathbf{p})}{f(\mathbf{p})}{e}^{ȷi\theta }{e}^{ȷ(k-i)\psi }{2}^{-k}\right|}^{1/k}\le \frac{D}{\mathrm{sep}(\mathbf{p},V(f))}.$$

$$\sum _{i=0}^k{\left(\left(\genfrac{}{}{0ex}{}{k}{i}\right)\left|\frac{f_{i,k-i}(\mathbf{p})}{f(\mathbf{p})}\right|e^{ȷ(k-i)\psi }{2}^{-k}\right)}^2=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }|P(\theta ){|}^{2}d\theta .$$

$$\sum _{i=0}^k{\left(\left(\genfrac{}{}{0ex}{}{k}{i}\right)\left|\frac{f_{i,k-i}(\mathbf{p})}{f(\mathbf{p})}\right|e^{ȷ(k-i)\psi }{2}^{-k}\right)}^2\le {\left(\frac{D}{\mathrm{sep}(\mathbf{p},V(f))}\right)}^{2k}.$$

$$\sum _{i=0}^k{\left(\left(\genfrac{}{}{0ex}{}{k}{i}\right)\left|\frac{f_{i,k-i}(\mathbf{p})}{f(\mathbf{p})}\right|e^{ȷ(k-i)\psi }{2}^{-k}\right)}^2>\underset{j=0,\dots ,k}{\max }{\left(\left|\frac{f_{j,k-j}(\mathbf{p})}{f(\mathbf{p})}\right|2^{-k}\right)}^2$$

$$\underset{i=0,\dots ,k}{\max }{\left|\frac{f_{i,k-i}(\mathbf{p})}{f(\mathbf{p})}\right|}^{1/k}<\frac{2D}{\mathrm{sep}(\mathbf{p},V(f))}.$$

$$|a_{0,0}|>\sum _{k\ge 1}\sum _{i=0}^k|a_{i,k-i}||x{|}^i|y{|}^{k-i}$$

$$\gamma\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;\max_{1\leq k\leq D}\max_{0\leq i\leq k}\left({\frac{k!}{\binom{k}{i}}\left|\frac{a_{i,k-i}}{a_{0}}\right|}\right)^{1/k},$$

$$1>\sum _{k\ge 1}\frac{\gamma }{k!}\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{k}{i}\right)|x{|}^i|y{|}^{k-i}=\sum _{k\ge 1}\frac{\gamma }{k!}(|x|+|y|{)}^k>\exp (\gamma \Vert \mathbf{x}{\Vert }_1)-1.$$

$$\gamma_{f}({\bf p})\;{\color[rgb]{0,0,1}\mathop{\mbox{\rm:=}}}\;\max_{1\leq k\leq D}\max_{0\leq i\leq k}{\left|\frac{f_{i,k-i}({\bf p})}{f({\bf p})}\right|}^{1/k},$$

$$\mathrm{sep}(\mathbf{p},V(f))\ge \frac{\ln 2}{\sqrt{2}\gamma (\mathbf{p})}\ge \frac{1}{3\gamma (\mathbf{p})}.$$

Full text

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The bound above can also be interpreted as an upper bound on the separation of a point ${\bf p}$ from the variety $V(f)$ in terms of the coefficients of the polynomial. Can a converse bound be given, i.e., a lower bound on the separation in terms of the coefficients. We next derive such a bound.

Suppose $f(0)\neq 0$, then we want to derive a lower bound on $\mathrm{sep}(0,V(f))$ in terms of the coefficients. Clearly, any ${\bf x}=(x,y)$ for which

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cannot be on the variety of $f$. Define

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where $D$ is the total degree. Then it follows that (15) is equivalent to

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Therefore, if ${\bf x}$ is such that $\|{\bf x}\|_{1}\gamma<\ln 2$ then $|f({\bf x})|>0$. In general, for any point ${\bf p}\in{\mathbb{C}}^{2}$ we can apply the argument above to the shifted polynomial to obtain the following: if

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then

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Besides their intrinsic interest, such bounds are useful in analyzing the complexity of certain algorithms. For instance, the bound given in (3) has been useful in bounding the running time of certain root isolation algorithms using the continuous amortization framework [Bur16, SB15]. We expect the generalization given above to be useful in deriving similar bounds on the running time of generalizations of corresponding algorithms that generally use subdivision (e.g., [PV04]).

Acknowledgement: The author is grateful to Chee Yap and Bernard Mourrain for their feedback on earlier drafts of the results presented here.