A Maple Implementation for Deterministically Certifying Isolated Simple Zeros of Over-Determined Polynomial Systems with Interval Arithmetic and Its Applications

Abstract

This paper presents a Maple implementation of an interval verification method for identifying isolated simple zeros in square polynomial systems. Compared to the known MATLAB (R2019b) implementation, the Maple-based approach achieves significantly higher numerical accuracy. The implementation enables polynomial evaluation at specific points to yield results with very small absolute values—sufficiently precise to reach error bounds computed through theoretical formulations for moderate-sized systems. This advancement allows the deterministic certification of isolated simple zeros in over-determined polynomial systems containing approximately 10,000 complex zeros. As a practical demonstration, the method is further applied to rigorously verify isolated multiple zeros in smaller-scale polynomial systems.

1. Introduction

Determining zeros of polynomial systems is a core computational task with critical applications in modern scientific fields. These include robotics for kinematic solutions [1] and machine learning for model training and inference [2]. Such applications frequently involve large-scale, over-determined systems where numerical certainty is paramount, underscoring the need for high-precision, certifiable verification methods.

A fundamental challenge in numerical algebraic computation is to determine whether a given point is indeed a zero of a polynomial system. Established methods such as Smale’s $\alpha$-theory [3] and interval verification techniques [4,5,6] provide deterministic certification for isolated simple zeros in square systems. While implementations like ‘alphaCertified’ [7] (based on $\alpha$-theory) and ‘INTLAB’ [8] (interval arithmetic in MATLAB) exist, neither directly extends to over-determined systems.

Dedieu and Shub generalized $\alpha$-theory to certify isolated simple zeros in over-determined systems [9], but their approach yields only a necessary condition for deterministic certification. The certified zero might coincide with a critical point of the system. Cheng and Dou addressed this by constructing an associated square system, proving a one-to-one correspondence between simple real zeros of both systems [10]. Their method achieves a sufficient and necessary certification condition if a function vanishes exactly at the given zero. However, verifying exact vanishing through purely numerical computation remains infeasible, as this itself constitutes another necessary condition. Alternative heuristic approaches [11,12,13] exist but certify zeros of perturbed systems rather than the original.

A symbolic-numeric hybrid method was proposed for verifying all simple zeros in zero-dimensional over-determined systems with rational coefficients [14]. This method employs exact rational univariate representations, enabling global verification through combined symbolic and numerical computations. Though challenging for large systems, it provides the first sufficient deterministic certification framework for isolated simple zeros in over-determined systems, to the authors’ knowledge.

Notably, for the foundational task of certifying individual solutions in square systems, recent work addresses the computational bottleneck of Smale’s $\alpha$-theory. By reformulating the $\alpha$-theory criteria using interval arithmetic—termed “$\alpha$-theory over regions”—Lee demonstrates a more efficient certification process that avoids costly exact arithmetic while maintaining rigor, offering a faster alternative to traditional implementations [15].

Beyond static verification, many core tasks in numerical algebraic geometry, such as computing monodromy actions, rely on certified path tracking (homotopy continuation). Guillemot and Lairez recently combined interval arithmetic with Taylor models to deliver a validated predictor-corrector scheme whose efficiency rivals non-certified implementations, advancing reliable computation in dynamic settings [16].

Extending certification to singular roots presents a significant challenge. For isolated multiple roots, deflation techniques transform the singularity into a regular root of an augmented system. Recent work by Mantzaflaris et al. introduces a certified Newton iteration on a deflated system that converges to both the exact singular root (of a slightly perturbed system) and its full multiplicity structure, using $\alpha$-theory for certification [17]. This approach highlights the demand for high-assurance methods in handling singular solutions.

This work develops a Maple implementation of a fixed point theory based interval arithmetic verification method, leveraging the interval arithmetic package ‘intpakX’ [18], along with some applications.

Although the toolbox ‘INTLAB’ is available for interval verification in MATLAB, an implementation in Maple remains of interest. The primary motivation lies in the limited numerical precision of MATLAB, which restricts the accuracy of the verified roots—a disadvantage for engineering problems requiring high precision. In contrast, Maple offers virtually unrestricted accuracy, enabling implementations to attain root-separation bounds for moderate-sized systems. This capability permits deterministic verification of whether a polynomial vanishes at an isolated zero (even when inexactly specified), thereby certifying isolated simple zeros in over-determined systems with up to 10,000 complex zeros.

As applications, the certification of isolated multiple zeros and their multiplicity structures is demonstrated in systems with real coefficients. Using deflation techniques [19,20,21,22,23,24,25,26], multiple zeros are transformed into simple zeros of over-determined systems, which can then be certified with the proposed method. Furthermore, this framework is extended to verify multiplicity structures of isolated singularities during their computation [22].

2. Main Results

In this section, some results of certifying simple zeros in square polynomial systems using interval arithmetic are presented.

Let $C^i\left(E\right)$ denote the class of all i-th order continuously differentiable functions defined on a vector space E. For $f\in C^k\left({\mathbb{R}}^n\right)$, define its gradient as $\nabla f=\left({\displaystyle \frac{\partial f}{\partial x_1}},\dots ,{\displaystyle \frac{\partial f}{\partial x_n}}\right)$. Given matrix A, denote its i-th row and column as $A_{i,:}$ and $A_{:,i}$ respectively.

Let $\mathbb{I}\mathbb{R}$, $\mathbb{I}{\mathbb{R}}^n$ and $\mathbb{I}{\mathbb{R}}^{n\times n}$ represent real intervals, real interval vectors and interval matrices. For interval X, $int\left(X\right)$ denotes its interior.

The standard interval verification method relies on the following key theorem [4,5,6].

Theorem 1.

Let $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with $F=(f_1,\dots ,f_n)\in C^1\left({\mathbb{R}}^n\right)$, and let $\tilde{x}\in {\mathbb{R}}^n,X\in \mathbb{I}{\mathbb{R}}^n$ with $0\in X$, and $R\in {\mathbb{R}}^{n\times n}$ be given. Suppose $M\in \mathbb{I}{\mathbb{R}}^{n\times n}$ satisfies:

$$\{\nabla f_i\left(x\right):x\in \tilde{x}+X\}\subset M_{i,:}.$$

(1)

Let I be the $n\times n$ identity matrix. If

$$-RF\left(\tilde{x}\right)+(I-RM)X\subset int\left(X\right),$$

(2)

then there exists a unique zero $\widehat{x}\in \tilde{x}+X$ satisfying $F\left(\widehat{x}\right)=0$. Furthermore, any matrix $\tilde{M}\in M$ is nonsingular, implying the Jacobian $J_F\left(\widehat{x}\right)$ is nonsingular.

For $F:{\mathbb{C}}^n\to {\mathbb{C}}^n$ with $F\in C^1\left({\mathbb{C}}^n\right)$, define an approximate zero $p\in {\mathbb{C}}^n$ of F as a point that quadratically converges to a true zero $\eta$ (i.e., $F\left(\eta \right)=0$) under Newton iteration.

Based on Theorem 1, Algorithm 1 is formulated and implemented in Maple through the function $\mathbf{verifysys}$ to realize the interval verification method. Consider a polynomial system $F=\{f_1,\dots ,f_n\}\subset \mathbb{C}[x_1,\dots ,x_n]$ and an approximate zero $\tilde{x}$. The procedure first refines $\tilde{x}$ to a higher-accuracy point p using Newton’s method, implemented via the function $\mathbf{newtonmethod}$. Subsequently, the Condition (2) is tested. A typical strategy to Satisfy (2) involves selecting an approximation R of the inverse Jacobian $J_F\left(p\right)$ and an interval vector X with small width containing zero. The corresponding algorithm is presented below.

Algorithm 1 Interval Verification Procedure

$\mathbf{Procedure}X,p=\mathbf{verifysys}(F,vars,\tilde{x},m):$ $\mathbf{Input}$: A square system $F\subset \mathbb{C}[x_1,\dots ,x_n]$, an approximate zero $\tilde{x}$ with numeric precision digits $m^{\prime }$, the verification digits $m(m^{\prime }<m)$, the set of variables $vars:=(x_1,\dots ,x_n)$. $\mathbf{Output}$: The verified interval X containing a unique zero of F and the refined zero p. $X=\varnothing$ means the verification fails. 1.  $p:=\mathbf{newtonmethod}(F,vars,\tilde{x},m)$. 2.  Set $X_0:=([-{10}^{-m},{10}^{-m}],\dots ,[-{10}^{-m},{10}^{-m}])\in \mathbb{I}{\mathbb{R}}^n$. 3.  Compute $R:={\left(J_F\left(p\right)\right)}^{-1}$, $M:=J_F(p+X_0)$. 4.  Compute $X_1:=-RF\left(p\right)+(I-RM)X_0$. (a)    If $X_1\subset int\left(X_0\right)$, let $X:=p+X_0$. (b)    Else, let $X:=\varnothing$. 5.  $\mathbf{Return}$: $X,p$.

Remark 1.

The verification precision m determines output digit accuracy. In practice, if required accuracy of the output results is m, take $m+m_1$ digits in the computation for some integer $m_1>0$. Here $m_1$ is chosen empirically.

Consider the following example for illustration.

Example 1.

Consider the polynomial system $F=\{x^2+y^2-4,x-y\}$ with an approximate root $p_0=(1.4,1.4)$. Let $vars:=[x,y]$ denote the system variables. Setting the verification precision to $m=10$ digits and employing $m+m_1=15$ working digits for computations, execute the algorithm verifysys to obtain:

$>X,p:=\mathit{verifysys}(F,vars,p_0,m):$

$>X$

$$\left[\begin{array}{c}[1.41421356227308,1.41421356247310]\\ [1.41421356227308,1.41421356247310]\end{array}\right]$$

$>p$

$$[1.41421356237309,1.41421356237309].$$

Here X represents certified interval enclosures containing the unique zero, and p denotes the refined root of F.

Remark 2.

While the theorem and our illustrative example are formulated over $\mathbb{R}$, the methodology remains valid for complex domains $\mathbb{C}$ (see [6]).

Remark 3.

The algorithm’s implementation is mainly based on interval arithmetic, thus, Maple’s package intpakX [27,28] is used to realize interval arithmetic. Users can download the package from [18] and initialize it prior to executing our software. This choice ensures rigorous interval computations with arbitrary precision.

Based on $\mathbf{verifysys}$, the following Algorithm 2 is presented, which gives a Maple function $\mathbf{issameroot}$ to determine whether two approximate zeros ${\tilde{x}}_1$ and ${\tilde{x}}_2$ of F correspond to a same zero or not.

Algorithm 2 Determining a Same Zero

$\mathbf{Procedure}A=\mathbf{issameroot}(F,vars,{\tilde{x}}_1,{\tilde{x}}_2,m):$ $\mathbf{Input}$: A square system $F\subset \mathbb{C}[x_1,\dots ,x_n]$, two approximate zeros ${\tilde{x}}_1$ and ${\tilde{x}}_2$, the verification digits m, the variables set $vars:=(x_1,\dots ,x_n)$. $\mathbf{Output}$: True, False or Unknown. 1.  $X_1:=\mathbf{verifysys}(F,vars,{\tilde{x}}_1,m)$. If $X_1=\varnothing$, $A:=\mathrm{Unknown}$. 2.  $X_2:=\mathbf{verifysys}(F,vars,{\tilde{x}}_2,m)$. If $X_2=\varnothing$, $A:=\mathrm{Unknown}$. 3.  If $X_1\ne \varnothing$ and $X_2\ne \varnothing$, do (a)  If $X_1\subset X_2$ or $X_2\subset X_1$, $A:=\mathrm{True}$. (b)  Else if $X_1\cap X_2=\varnothing$, $A:=\mathrm{False}$. (c)  Else, $m:=m+10$, go to Step (1). 4.  $\mathbf{Return}$: A.

Remark 4.

1. The termination of Algorithm 2 follows from Newton’s quadratic convergence and Maple’s unbounded precision arithmetic: iterative refinement eventually separates $X_1$ and $X_2$ theoretically, yielding either inclusion $X_1\subset X_2$ (or vice versa) or disjointness $X_1\cap X_2=\varnothing$.

2. The output “True” from issameroot indicates that ${\tilde{x}}_1$ and ${\tilde{x}}_2$ converge to the same zero, “False” indicates they do not, and “Unknown” indicates an inability to determine convergence at the current precision. To improve success, the verification precision for $X_2$ can be set to $m+5$, making the condition $X_2\subset X_1$ easier to satisfy. Furthermore, in step (c), the precision can be updated via $m:=m+a$, where the increment a may be dynamically adjusted according to the magnitude of m.

3. Some Applications

In this section, two applications of Algorithm 1 are presented. A key challenge involves determining whether a function vanishes at a given isolated zero of a system when the zero is not provided exactly. To address this, the methodology provided in this paper leverages Maple’s high-precision arithmetic to refine the system’s zeros. Specifically, if a function is theoretically zero at the given isolated zero, the refinement process ensures that its absolute value falls below a non-zero threshold derived from numerical precision bounds. This approach ultimately enables rigorous zero certification for over-determined polynomial systems.

The DMM bound [29] for polynomial system roots is introduced below.

Theorem 2.

Let $F=\{f_1,\dots ,f_n\}\subset \mathbb{Z}[x_1,\dots ,x_n]$ be a polynomial system where each $f_i$ satisfies $deg\left(f_i\right)\le d$ and $\mathcal{L}\left(f_i\right)\le \tau$ for $1\le i\le n$. Here $deg\left(f_i\right)$ denotes the maximum total degree of $f_i$, and $\mathcal{L}\left(f_i\right)$ represents the maximal coefficient bit-size of $f_i$. Let D be the number of isolated roots of F in ${\mathbb{C}}^n$, ordered as $0<|{\gamma }_1|<\dots <|{\gamma }_D|$, where ${\gamma }_{j,k}$ denotes the k-th coordinate of ${\gamma }_j$. Then for any k, either ${\gamma }_{j,k}=0$ or

$$2^{(n^2-n)log\left(\sqrt{d}\right)-d^n-n(\tau +nlog\left(d\right)+2)d^{n-1}}\le \left|{\gamma }_{j,k}\right|\le 2^{(n^2-n)log\left(\sqrt{d}\right)+d^n+n(\tau +nlog\left(d\right)+2)d^{n-1}}.$$

Let $b_F$ and $B_F$ denote the lower and upper bounds, respectively, i.e., $b_F\le \left|{\gamma }_{j,k}\right|\le B_F$. A method for certifying simple roots of over-determined systems can now be proposed.

3.1. Verifying a Zero of an Over-Determined Polynomial System

Let $\mathsf{\Sigma }=\{f_1,\dots ,f_N\}\subset \mathbb{Z}[x_1,\dots ,x_n](N>n)$ be an over-determined polynomial system, and let $\tilde{x}\in {\mathbb{C}}^n$ be an approximate simple zero of $\mathsf{\Sigma }$. Without loss of generality, assume there exist n polynomials $F=\{f_1,\dots ,f_n\}\subset \mathsf{\Sigma }$ for which $\tilde{x}$ serves as an approximate simple zero of F. Applying the function $\mathbf{verifysys}$ from Algorithm 1 to F with $\tilde{x}$ yields a verified interval X. For the remaining polynomials $\{f_{n+1},\dots ,f_N\}$, sequential verification is performed. For each $f_{n+i}(1\le i\le N-n)$, an auxiliary variable $y_i$ is introduced, constructing the square system

$$G_i=\{f_1,\dots ,f_n,y_i-f_{n+i}\},1\le i\le N-n.$$

The point $(\tilde{x},0)$ constitutes an approximate simple zero of $G_i$. Applying $\mathbf{verifysys}$ to $G_i$ with this initial approximation $(\tilde{x},0)$ produces a verified interval for $y_i$: $Y_i=[y_{i,1},y_{i,2}]$, and a verified interval for variables $x_1,\dots ,x_n$: $X_i$.

Theorem 1 makes sure that a unique zero $\widehat{x}\in X_i$ and $\widehat{y_i}\in Y_i$ exist satisfying $G_i(x,y_i)=0$. Using Maple’s high-precision arithmetic, $Y_i$ is refined such that $-b_{G_i}<y_{i,1}<y_{i,2}<b_{G_i}$, where $b_{G_i}$ is the DMM lower bound for $G_i$. Theorem 2 yields

$$\mid \widehat{y_i}\mid \le max\{\mid y_{i,1}\mid ,\mid y_{i,2}\mid \}<\mid b_{G_i}\mid ,$$

which determines that $\widehat{y_i}=0$. Therefore, if $0\in Y_i$, Theorems 1 and 2 guarantee $y_i=0$, thereby certifying that the over-determined system $F,f_{n+i}$ has a unique common zero in X. Iterating this process for all $1\le i\le N-n$, and confirming $X_i\subset X$ at each step, ultimately establishes that the over-determined system $\mathsf{\Sigma }$ possesses a certified simple zero in X.

Based on the above analysis, Algorithm 3 is formally described below, which corresponds to the Maple function verifyodsys.

Algorithm 3 Verifying Zeros of Over-determined Systems

$\mathbf{Procedure}X=\mathbf{verifyodsys}(\mathsf{\Sigma },vars,\tilde{x},m_0):$ $\mathbf{Input}$: An over-determined system $\mathsf{\Sigma }\subset \mathbb{Z}[x_1,\dots ,x_n]$, an approximate simple zero $\tilde{x}$, the initial accuracy $m_0$ and the variables set $vars:=(x_1,\dots ,x_n)$. $\mathbf{Output}$: The verified interval X contained a unique root of $\mathsf{\Sigma }$. $X=\varnothing$ means the verification fails. 1.  Select a square system F from $\mathsf{\Sigma }$ such that $\tilde{x}$ is an approximate simple root of F. Set $G:=\mathsf{\Sigma }\setminus F$. 2.  $X,p:=\mathbf{verifysys}(F,vars,\tilde{x},m_0)$. 3.  Set $newvars:=(vars,y)$. 4.  While $X\ne \varnothing$ and $G\ne \varnothing$ do (a)    Select a polynomial $g\in G$ and delete it from G. (b)    Compute the DMM bound $b_H$ of the system $H:=\{F,y-g\}$. (c)    Set the verification digits $m:=ceil\left(\right|log\left(b_H\right)\left|\right)+1$ and let the computing digits be $m+50$. (d)    $E:=\mathbf{verifysys}(H,newvars,(p,0),m)$. Let $X^{\prime }:=E(1..n),Y:=E(n+1)$. (e)    If $0\notin Y$ or $X^{\prime }\not\subset X$, set $X:=\varnothing$. 5.  $\mathbf{Return}$: X.

Remark 5.

1. The initial accuracy $m_0$ determines output digit accuracy and its value can be selected in the specific calculation by combining the value of Maple’s calculation effective environment Digits.

2. By Theorem 2, the dominant term $d^n$ implies a double-exponential dependency on the number of variables n and the degree d, making direct computation prohibitive for large n or moderate d. This dependency limits the method’s applicability. Consequently, the presented algorithm is currently suited only for small-scale systems—for instance, when n is small and d is large, when d is small and n is large, or when neither parameter is particularly large.

Consider the following example for illustration.

Example 2.

Consider the over-determined system $\mathsf{\Sigma }=\{x^2+y^2-2,y-x^3,y^2-x\}$ with an approximate zero $p_0=(1.002,0.993)$. Let $vars:=[x,y]$ denote the system variables. Take initial accuracy $m_0=15$. Execute the algorithm verifyodsys,

1.

Select a square system $F=\{x^2+y^2-2,y-x^3\}$ from Σ, where $p_0$ serves as an approximate simple root. Set residual system $G:=\{y^2-x\}$.

2.

Execute verification: $X,p:=\mathit{verifysys}(F,vars,\tilde{x},m_0)$, yielding:

$X:=$

$$\left[\begin{array}{c}[0.99999999999999899999,1.0000000000000010001]\\ [0.99999999999999899999,1.0000000000000010001]\end{array}\right]$$

and $p:=[1.0000000000000000000,1.0000000000000000000]$.

3.

Extend variables: $newvars:=(vars,y)$ and let $g:=y^2-x$.

4.

Compute DMM bound $b_H$ for augmented system $H:=\{F,y-g\}$, obtaining $b_H=2^{-78log\left(3\right)-135}$.

5.

Set precision parameter: $m:=68$ with computation digits 118.

6.

Perform verification: $E:=\mathit{verifysys}(H,newvars,(p,0),m)$. Extract $X^{\prime }:=E(1..n)$ and $Y:=E(n+1)$.

7.

Certification: since $0\in Y$ and $X^{\prime }\subset X$, the interval X is validated.

Final verification result:

$>X:=\mathit{verifyodsys}(\mathsf{\Sigma },vars,p_0,m_0):$

$>X$

$$\left[\begin{array}{c}[0.99999999999999899999,1.0000000000000010001]\\ [0.99999999999999899999,1.0000000000000010001]\end{array}\right]$$

This certifies that the over-determined system Σ possesses a unique root in X.

3.2. Verifying Multiple Zeros of Polynomial Systems

Verifying multiple zeros of polynomial systems presents significant challenges, as arbitrary small coefficient perturbations may transform a multiple root into a cluster of simple roots or eliminate it entirely. Deflation techniques offer a principal approach to address this issue by transforming multiple roots of the original system into simple roots of an augmented system. While effective, this methodology typically increases the system size and may introduce perturbations in certain implementations. For deterministic certification, the derived system must preserve the original system’s zeros without perturbation when applying deflation techniques. Following established methods [20,21,22,23,24,25,26,30], over-determined systems are constructed containing the original system, with or without introducing new variables. Utilizing the certification framework from Section 3, these over-determined systems and their associated multiple zeros can subsequently be validated, as demonstrated below.

Example 3

([22]). Consider $F=\{f_1,f_2\}=\{x+y^2,x^2+y^2\}$ with an approximate multiple zero $p_0=(0.00013,0.00091)$. Let $g={\displaystyle \frac{\partial f_2}{\partial y}}=2y$. Remarkably, $p_0=(0.00013,0.00091)$ serves as an approximate simple zero for the augmented system $G=\{f_1,g\}$. We certify the over-determined system $\mathsf{\Sigma }=F\cup G$ with $\tilde{p}$. Set $vars:=[x,y]$ and the initial accuracy $m_0=15$.

$>X:=\mathit{verifyodsys}(\mathsf{\Sigma },vars,p_0,m_0):$

$>X$

$$\left[\begin{array}{c}[-1.0000000000000000001·{10}^{-15},1.0000000000000000001·{10}^{-15}]\\ [-1.0000000000000000001·{10}^{-15},1.0000000000000000001·{10}^{-15}]\end{array}\right]$$

That is to say, the over-determined system Σ possesses a unique root in X, in other words, the system F possesses a unique multiple root in X.

In [22], a new deflation method for isolated singular zeros is introduced, along with a construction to compute multiplicity structures. Based on computed dual bases at singular zeros with an original system, an over-determined system containing the information of multiplicity structures can be derived. Certifying this over-determined system at the zeros provides deterministic verification of the multiplicity structures of singular zeros of the original system. The following example illustrates this process.

Example 4.

Extending Example 3, the method from [22] is used to compute the deflated system $G=\{x+y^2,\mu +2y,x^2+y^2,2\mu x+2y\}$, which includes the introduced variable μ. The multiplicity structure at $\tilde{p}$ for F is determined to be $\{1,{\partial }_2+\mu {\partial }_1\}$. The system G has an approximate simple zero at $p_0=(x,y,\mu )=(0.00013,0.00091,0.1)$. The algorithm $\mathit{verifyodsys}$ is applied to verify G with the initial point $p_0$, thereby certifying the multiplicity structure simultaneously. Set $vars:=[x,y,\mu ]$ and the initial accuracy $m_0=15$.

$>X:=\mathit{verifyodsys}(G,vars,p_0,m_0):$

$>X$

$$\left[\begin{array}{c}[-1.0000000000000000001·{10}^{-15},1.0000000000000000001·{10}^{-15}]\\ [-1.0000000000000000001·{10}^{-15},1.0000000000000000001·{10}^{-15}]\\ [-1.0000000000000000001·{10}^{-15},1.0000000000000000001·{10}^{-15}]\end{array}\right]$$

Thus, the multiplicity structure $\{1,{\partial }_2+\mu {\partial }_1\}$ of the system can be verified at the given zero, where μ is a real number contained within the last coordinate interval of X. Furthermore, the DMM bound can be applied to establish a bound for μ in this example, ultimately determining that $\mu =0$, i.e., the multiplicity structure of the system at the given zero is $1,{\partial }_2$.

4. Examples

To demonstrate the applicability of Algorithm 3, the main program $\mathbf{verifyodsys}$ is tested with several large systems.

For computational evaluation, polynomial systems with different degrees, terms, and coefficients are randomly generated in Maple. First, a square polynomial system $\{g_1,\dots ,g_n\}\subset \mathbb{Z}[x_1,\dots ,x_n]$ with low degrees is generated, and an approximate root $p_0$ is computed. Subsequently, a set of polynomials $\{h_1,\dots ,h_N\}\subset \mathbb{Z}[x_1,\dots ,x_n]$ with $N>n$ and high degrees is generated. Let $f_i=g_i{h}_i$ for $1\le i\le n$, and $f_j=g_{k_j}{h}_j$ for $n+1\le j\le N$, where $k_j$ is randomly selected from $\{1,2,\dots ,n\}$. This typically yields an over-determined polynomial system $F=\{f_1,\dots ,f_N\}$ with an isolated simple zero $p_0$. Denote by d the maximal total degree of $f_i$, and by $\left|c\right|$ the maximal absolute coefficient of $f_i$. Several systems of this kind—labeled ex1, ex2, ex3, ex4 and ex5—are constructed and tested with the implementation. Prior to run our code, all polynomials are expanded. Results are presented in

Table 1. Performance of Algorithm 3 for Large Systems.

System	n	N	d	$\left|\mathit{c}\right|\le$	Verified Digits	Times
ex1	2	4	53	10	95,323	235.078 s
ex2	2	3	53	10	95,323	111.532 s
ex3	2	3	80	10	270,564	696.045 s
ex4	3	4	20	10	231,022	906.179 s
ex5	4	5	10	10	308,744	1535.081 s

. It should be noted that these randomly generated high-degree polynomials are primarily used as benchmark problems to test the performance of algorithms, but systems of similar algebraic structure do arise in applications such as robot arm kinematics and workspace analysis.

The examples are in the following:

ex1. 

$f_1=(3x_1^2{x}_2+x_2^3-3x_1{x}_2+2x_2^2-3x_1)(-2x_1^{50}-3x_1{x}_2^{45}+3x_1^{31}{x}_2^3+2x_1^9{x}_2^{14}-3x_1^2$$-2x_1{x}_2+3x_2^2+2x_1-2x_2+3)$,

$f_2=(-2x_1^2{x}_2-2x_2^3-x_1{x}_2-2x_2^2-3x_1+4)(-x_1^{32}{x}_2^{13}+2x_1^{32}{x}_2^{12}-2x_1^2{x}_2^{41}+2x_1^{21}{x}_2^{12}$$-3x_1^2-3x_1{x}_2+3x_2^2+x_1-2x_2+1)$,

$f_3=(3x_1^2{x}_2+x_2^3-3x_1{x}_2+2x_2^2-3x_1)(-x_1^{35}{x}_2+2x_1^{12}{x}_2^{23}+x_1^{32}{x}_2-3x_1^{10}{x}_2^{22}+3x_1^{11}{x}_2^2$$+x_1{x}_2+2x_1+3x_2+3)$,

$f_4=(-2x_1^2{x}_2-2x_2^3-x_1{x}_2-2x_2^2-3x_1+4)(-2x_1^{45}{x}_2^5+2x_1^{21}{x}_2^{12}+3x_1^8{x}_2^{19}+2x_1^{21}{x}_2^3$$+2x_1^{19}{x}_2+2x_2^{13}+2x_1^2-3x_1{x}_2+x_2^2+x_1-2x_2-1)$,

$\tilde{x}=(0.3954,0.7927)$.

ex2. 

Take $f_1,f_2,f_3$ in ex 1, $\tilde{x}=(0.3954,0.7927)$.

ex3. 

$f_1=(3x_1{x}_2^2+2x_2^3+2x_1{x}_2+3x_1-2)(-2x_1^{77}-2x_1^{54}{x}_2^{20}-2x_1^{63}{x}_2^8-3x_1^{54}{x}_2^4-x_1^{17}{x}_2^{38}$$+x_1^{16}{x}_2^{20}-x_1^2+2x_1{x}_2+2x_2^2+x_1+x_2-3)$,

$f_2=(x_1^3-2x_1{x}_2^2-2x_2^3-2x_1^2-x_2^2+3x_1)(-3x_1^{54}{x}_2^{22}+2x_1^7{x}_2^{61}-x_1^{27}{x}_2^{14}-3x_1^{10}{x}_2^{18}$$-x_1^2-2x_1{x}_2-3x_2^3-2x_1-3x_2)$,

$f_3=(3x_1{x}_2^2+2x_2^3+2x_1{x}_2+3x_1-2)(-x_1^{45}{x}_2^{31}-3x_1^{41}{x}_2^{33}-3x_1^{28}{x}_2^{45}-x_2^{13}-2x_1{x}_2$$-2x_2^2+x_2)$,

$\tilde{x}=(0.3364,0.5451)$.

ex4. 

$f_1=(-2x_1^2{x}_2+3x_1{x}_2{x}_3-x_2^3+3x_2{x}_3^2+x_1^2-1)(-3x_1{x}_2^3{x}_3^{13}-3x_1{x}_3^{16}+x_1^{12}{x}_2^2{x}_3^2$$-x_1^6{x}_2^5{x}_3^5+x_1^3{x}_2{x}_3^9+2x_1^8{x}_3^4-2x_1{x}_2-2x_1{x}_3-x_2{x}_3-x_3^3+1)$,

$f_2=(2x_1{x}_3^2+2x_2^2{x}_3+2x_2{x}_3^2-2x_3^3+3x_1^2+x_1+2)(-3x_1^{13}{x}_2^2{x}_3^2-3x_1^9{x}_2^2{x}_3^5-2x_1^6{x}_2^3{x}_3^5$$-3x_1^4{x}_3^7-x_1{x}_2{x}_3^5+2x_1{x}_2-3x_2^2+3x_2{x}_3-x_3^2-2x_1+3x_3)$,

$f_3=(-2x_1^2{x}_2+3x_1{x}_3^2-x_1{x}_3+2x_2^2-2)(x_1^2{x}_2^{13}{x}_3-2x_1^2{x}_2^7{x}_3^7+3x_1^9{x}_2^2{x}_3^4+x_1{x}_2^{13}$$-3x_1^{11}{x}_2{x}_3+3x_1^5{x}_2^3-2x_1{x}_2+2x_1{x}_3+x_2{x}_3-2x_1-2)$,

$f_4=(-2x_1^2{x}_2+3x_1{x}_3^2-x_1{x}_3+2x_2^2-2)(x_1^7{x}_2{x}_3^9-3x_1{x}_2^8{x}_3^8-3x_1^6{x}_2^7{x}_3^3+3x_1^4{x}_2{x}_3^{11}$$-x_1^2{x}_2{x}_3^5-x_1{x}_2+2x_3^2-2x_1+x_2+2x_3)$,

$\tilde{x}=(-0.5492,-1.0701,0.9384)$.

ex5. 

$f_1=(2x_2{x}_3{x}_4+3x_3^2{x}_4-x_1{x}_2+x_4^2-x_4+2)(3x_1{x}_2^6+2x_2^3{x}_4^4-x_2^2{x}_3^4{x}_4+x_2^5{x}_4$$+x_2{x}_3{x}_4^4+3x_1{x}_2+3x_2^2+3x_2{x}_3-3x_2{x}_4+2x_1+3x_2-x_4)$,

$f_2=(2x_1{x}_2{x}_4-2x_4^3+x_1{x}_2-3x_2+3)(3x_1^4{x}_2^3-x_1^2{x}_3{x}_4^4+x_1{x}_2{x}_3^4{x}_4-2x_1^2{x}_2^3{x}_3-x_2^2{x}_3^3$$+x_1{x}_3{x}_4-x_1{x}_4-3x_2{x}_3-3x_3{x}_4-x_4^2+3x_1)$,

$f_3=(2x_2{x}_3^2-2x_3^3+3x_2^2-3x_2+1)(x_1^5{x}_2^2+x_2^4{x}_3^3+x_2^4{x}_4^3-3x_1^3{x}_3^2{x}_4-3x_1{x}_2{x}_3-x_2^2$$-x_3+2x_4+2)$,

$f_4=(-3x_1^2{x}_2+3x_1^2{x}_3-2x_1{x}_2{x}_4-3x_2^3-2x_4^3-x_4+2)(3x_1^5{x}_3^2-3x_1^2{x}_2{x}_3^3{x}_4+3x_1{x}_3^4{x}_4^2$$+3x_1^2{x}_2{x}_3{x}_4+2x_2^2{x}_3^2{x}_4+3x_1{x}_2-3x_1{x}_4-x_2^2+2x_2{x}_3-x_4^2-2x_2-3)$,

$f_5=(2x_2{x}_3{x}_4+3x_3^2{x}_4-x_1{x}_2+x_4^2-x_4+2)(2x_1{x}_3^2{x}_4^4-3x_3^7-2x_1^2{x}_2^3+3x_1^2-2x_1{x}_2-2x_1{x}_3$$+x_1{x}_4+2x_1+2x_3)$,

$\tilde{x}=(-1.8760,1.1121,1.4423,-0.51515)$.

Several examples of systems F with approximate multiple zeros $p_0$ are also tested. These examples are sourced from the paper [20]. Each system F is transformed into a new system G that possesses an approximate simple zero $p_0$. The over-determined system $F\cup G$ is then verified using the algorithm $\mathbf{verifyodsys}$ with $p_0$. The results are presented in

Table 2. Performance of Algorithm 3 for Small Systems.

System	n	N	d	$\left|\mathit{c}\right|\le$	Verified Digits	Times
DZ1	4	8	4	1	4131	1.669 s
DZ2	3	5	4	8	965	0.062 s
cbms1	3	6	3	1	263	0.063 s
cbms2	3	6	3	3	328	2.808 s
Decker2	2	3	4	1	122	0.047 s

.

The examples are in the following:

• DZ1: $F=\{x_1^4-x_2{x}_3{x}_4,x_2^4-x_1{x}_3{x}_4,x_3^4-x_1{x}_2{x}_4,x_4^4-x_1{x}_2{x}_3\},\tilde{x}=(0.000081,0.000091,0.000013,0.000091),G=\{-x_2,-x_3,-x_4,-x_1\}$.
• DZ2: $F=\{x_1^4,x_1^2{x}_2+x_2^4,-7x_1^3-8x_1^2+x_3^2+x_3\},\tilde{x}=(0.000063,0.00001,-0.999972),$$G=\{-7x_1^3-8x_1^2+x_3^2+x_3,4x_1,x_2\}$.
• cbms1: $F=\{x_1^3-x_2{x}_3,x_2^3-x_1{x}_3,x_3^3-x_1{x}_2\},\tilde{x}=(0.000016,0.000097,0.000096),$$G=\{-x_2,-x_3,-x_1\}$.
• cbms2: $F=\{x_1^3-3x_1^2{x}_2+3x_1{x}_2^2-x_2^3-x_3^2,-x_1^2+x_2^3-3x_2^2{x}_3+3x_2{x}_3^2-x_3^3,-x_1^3$$+3x_1^2{x}_3-3x_1{x}_3^2-x_2^2+x_3^3\},\tilde{x}=(0.000092,0.000079,0.000096),G=\{-2x_3,-2x_1,-2x_2\}$.
• Decker2: $F=x_2^3+x_1,x_1^2{x}_2-x_2^4,\tilde{x}=(0.000014,0.000042),G=\{x_2,x_2^3+x_1\}$.

Table 1 and Table 2 demonstrate the algorithm’s effectiveness for both large-scale and small-scale systems. Specifically, Table 1 shows that deterministic certification of isolated simple zeros in over-determined polynomial systems containing approximately 10,000 complex zeros, as in example ex5, is achievable. Furthermore, Table 2 illustrates that Algorithm 3 can verify singular roots of the original system by certifying the simple roots of the derived over-determined system.

Finally, to illustrate the advantages of the interval verification algorithm implemented in Maple, a comparison is made with ‘INTLAB’ in MATLAB. The results are presented in

Table 3. Comparison of $\mathbf{INTLAB}$ with $\mathbf{verifyodsys}$.

System	n	N	d	INTLAB		Verifyodsys
				Width	Times	Width	Times
ex_3	2	2	80	$1.45\times {10}^{-14}$	0.58	$2\times {10}^{-15}$	0.062
ex_4	3	3	20	$2.26\times {10}^{-14}$	1.34	$2\times {10}^{-15}$	0.172
ex_5	4	4	10	$3.98\times {10}^{-13}$	1.46	$2\times {10}^{-15}$	0.235
G_DZ2	3	3	3	$4.44\times {10}^{-16}$	0.11	$2\times {10}^{-15}$	0.031
G_Decker2	2	2	3	$1.98\times {10}^{-323}$	0.08	$2\times {10}^{-15}$	0.016

, where $\mathbf{verifyodsys}$ is executed with $\mathrm{Digits}=50$. The examples ex_3, ex_4, and ex_5 correspond to ex3, ex4, and ex5 from the previous discussion. For each example, only the first few necessary polynomials are selected to form a square system. G_DZ2 and G_Decker2 are derived from the system G in DZ2 and Decker2. The $Width$ parameter specifies the maximum interval width of all verified roots. Table 3 clearly shows that the proposed algorithm achieves a smaller interval width for ex_3, ex_4, and ex_5, along with a shorter time for all systems. Although the method yields a larger interval width for G_DZ2 and G_Decker2, increasing the Digits setting and the initial verification accuracy $m_0$ leads to a smaller width. This flexibility stems from Maple’s advantage of virtually unlimited precision. For instance, with $\mathrm{Digits}=500$ and $m_0=500$, a width $2.0\times {10}^{-500}$ is obtained for G_Decker2.

5. Conclusions

This paper first develops an interval verification method in Maple, named $\mathbf{verifysys}$, for certifying isolated simple zeros of square polynomial systems. Building on this foundation, a Maple function $\mathbf{issameroot}$ is introduced to deterministically validate whether two approximate zeros of a system correspond to the same underlying zero.

Subsequently, a method is presented for verifying isolated simple zeros of over-determined systems. This involves introducing a parametric variable and determining whether it is zero based on the DMM bound. The main idea is implemented in Maple as the function $\mathbf{verifyodsys}$. Finally, the algorithm $\mathbf{verifyodsys}$ is applied to verify isolated multiple zeros of a system by using deflation techniques to derive an over-determined system from the original system.

As noted in Remark 5, the computational cost of evaluating the DMM bound limits the presented method’s applicability. Therefore, developing an efficient algorithm to compute the DMM bound, or identifying a more tractable local bound that can serve as an effective substitute, constitutes a key direction for future research on this problem.