# Direct Algorithm for Bernstein Enclosure Boundary of Polynomials

Authors: Tareq Hamadneh; Hassan Al Zoubi; Ibraheem Abu Falahah; Mutaz Al-Sabbagh
Journal: Journal of Mathematics
 (2022)

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## Abstract

Multivariate polynomials of finite degree can be expanded into Bernstein form over a given simplex domain. The minimum and maximum Bernstein control points optimize the polynomial curve over the same domain. In this paper, we address methods for computing these control points in the simplicial case of maximum degreeL. To this end, we provide arithmetic operations and properties for obtaining a fast computational method of Bernstein coefficients. Furthermore, we give an algorithm for direct determination of the minimum and maximum Bernstein coefficients (enclosure boundary) in the simplicial multivariate case. Subsequently, the implicit form, monotonicity, and dominance cases are investigated.


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## Body

## 1. Introduction

The enclosure of Bernstein functionF estimates the range of polynomials over a given simplex. In order to determine the enclosure boundary, all Bernstein coefficients of degree L are needed in the traditional approach and their number is large for Bernstein functions with moderately many variables. The problem of minimizing and bounding polynomials for global optimization problems was considered in [1]. The use of Bernstein expansion for a given power form polynomial over a simplex (triangles) was considered in [2–6]. Applications of a similar approach on shape designs and geometric representations in computer-aided geometric design were generalized in [7]. Furthermore, a computational method for reaction diffusion model was addressed in [8]. In [9], the authors published results in degree elevation and subdivision of the underlying simplex of Bernstein basis for solving global optimization and system problems. In [10, 11], the tensorial Bernstein case over boxes was addressed for computing the enclosure range of a given (multivariate) rational polynomial function, which is slow. Generally speaking, minimizing and maximizing of Bernstein coefficients provide bounds for the range of its polynomial function F over any given simplex, whereas the complexity of computing these coefficients is high. In this paper, we simplify the computational of Bernstein coefficients in high degree over a multidimensional simplex. Moreover, we provide a fast method for direct determination of the Bernstein enclosure boundary depending on the indices. This method covers the monotonicity of indices, tolerance case, and implicit Bernstein coefficients. Since our results are mathematically proven, we only give simple examples to be followed by readers.This paper is organized as follows. In the next section, we briefly recall the simplical polynomial Bernstein form. In Section3, we present the Bernstein expansion and properties over a simplex. The main results about the optimization and fast computation of Bernstein control points are given in Section 4. Conclusions are given in Section 5.


## 2. Background of Bernstein Expansion

We present the background and fundamental notations of Bernstein basis over a nondegenerate simplex. From the literature, we give the following definition of simplices.Definition 1.
Letσ0,…,σn be n+1 points of ℝn. The list v=σ0,…,σn defines a simplex of vertices σ0,…,σn. The convex hull of the vertex points σ0,…,σn of v is the set of ℝn defined as v. The largest edge of v defines the diameter of v.Throughout this work, the pointsσ0,…,σn are affinely independent in which case the simplex v is nondegenerate. For simplicity, we consider the standard simplex Δ=e0,e1,…,en, where e0 is the zero vector in ℝn and ei is the ith vector of the canonical basis of ℝn, i∈1,…,n. This is because any simplex v in ℝn can be linearly transformed to Δ. We also recall that x∈ℝn can be formulated as an affine combination of σ0,…,σn with the barycentric coordinates χ0,…,χn. If x=x1,…,xn∈Δ, then χ1,…,χn=x1,…,xn and χ0=1−∑i=1nxi. For every multi-index α=α0,…,αn∈ℕn+1 and χ=χ0,…,χn∈ℝn+1, we write α=α0+⋯+αn and χα≔∏i=0nχiαi. Let the entry-wise relation ≤ be given, then for α,β∈ℕn+1 with β≤α, we define(1)αβ≔∏i=0nαiβi.IfL∈ℕ is the degree of any polynomial function such that α=L, then we use the notation Lα=L!/α0!⋯αn!.Definition 2.
The Bernstein basis of maximum degreeL over v is defined as SαLα=L, where(2)SαL=Lαχα.Note that Bernstein basis takes nonnegative values onv and ∑α=LSαL=1. Let F be a power form polynomial of degree l,(3)Fx=∑β≤lcβxβ.Since the Bernstein expansion forms a basis of the vector spaceℝnX of polynomials of maximum degree L, see Proposition 1.6 in [7], then Fx can be expanded as l≤L(4)Fx=∑α=LCαF,L,vSαLx,where CαF,L,v denote the simplicial coefficients of Bernstein polynomial F of degree L over v.Remark 1.
LetΔ=v, the grid points of degree L associated to Δ are the points(5)σαL,Δ=α0e0+⋯+αnenL∈ℝnα=L,where the associated control points to F are(6)σαL,Δ,CαF,L,Δ∈ℝn+1α=L.Proposition 1 (see Proposition 2.7 [9]).
ForF∈ℝnX, the following properties hold:(1)
Interpolation at the vertices is as follows:(7)CLei=Fσi,i=0,…,n;(2)
Convex hull of control points: the graph ofF over v is convexly optimized by the convex hull of its associated control points, see Figure 1.(3)
Enclosure bound property: the following bound forF holds:(8)minα=LCαF,L,v≤Fx≤maxα=LCαF,L,v,x∈v.Figure 1
The curve of a univariate polynomialp, the colored convex hull of control points optimizes the polynomial curve, and the minimum Bernstein value approximates the minimum range.Theorem 1 (see Theorem 3.3 in [12]).
LetFx be a polynomial in Bernstein basis of degree L. Then, its power form is(9)Fx=∑α=Lcαxα,where(10)cα=∑β=0α−1α−βLααβCβL,α=L.


## 3. Simplification of Bernstein

First, we provide the method of affine transformation ofv upon the standard simplex Δ, where the barycentric coordinates χ can be transformed to Cartesian coordinates. Subsequently, we simply express power form polynomials into Bernstein form over simplices. Let v=σ0,…,σn and σj=x1j,…,xnj for all j∈0,…,n. Therefore, x=x1,…,xn=χ1σ1−σ0+⋯+χnσn−σ0+σ0 and x1=χ1x11−x10+⋯+χnx1n−x10+x10,…,xn=χ1xn1−xn0+⋯+χnxnn−xn0+xn0. Then, we have(11)x1⋮xn=x11−x10…x1n−x10⋮⋮xn1−xn0…xnn−xn0.χ1⋮χn+x10⋮xn0,from which we have(12)χ1⋮χn=x11−x10…x1n−x10⋮⋮xn1−xn0…xnn−xn0−1.x1⋮xn−x10⋮xn0.In the following, we simplify the method of expanding power form polynomials in Bernstein form over a simplex.Notation 1.
Considerα=α^,α0∈ℕn+1 with χ∈ℝn+1, and from (3), we get 0≤β1≤l≕l1,0≤β2≤l−β1≕l2,…,0≤β^n≤l−β1+⋯+βn−1≕ln. Any polynomial power form Fx in (3) of degree l^=l1,…,ln can be expressed on Δ as(13)Fx=∑β^≤l^cβ^xβ^.
From Notation1, we can write ∑β^≤l=∑β1≤l1⋯∑βn≤ln. Additionally α^+α0=L,(14)Lα^,α0=Lα1.L−α1α2…L−α1+⋯+αn−1αn.L−α1+⋯+αnα0≕L1α1…Lnαn.L0α0≕L^α^,α0.Proposition 2.
Forα^,α0∈ℕn+1 and L^∈ℕn+1, the simplicial Bernstein expansion of maximum degree L can be given as(15)Fx=∑α^,α0≤L^Cα^,α0L^FSα^,α0L^x,where(16)Sα^,α0L^x=L^α^,α0xα^1−xα0,(17)Cα^,α0L^F=∑β^≤α^α^β^L^β^,β0cβ^,0≤α^,α0≤L^.Proof.
LetF be a polynomial power form of maximum degree l. For β0=L−β^, we have L≥l(18)Fx=∑β^≤lcβ^xβ^=∑β1=0l1⋯∑βn=0lncβ^x1β1⋯xnβn=∑β1≤l1⋯∑βn≤lncβ^x1β1⋯xnβnx+1−xL−β^=∑β1≤l1⋯∑βn≤lncβ^x1β1⋯xnβn∑γ^≤L−β^L−β^γ^xγ^1−xL−β^−γ^=∑β^≤l^∑γ^≤L−β^aβ^L−β^γ^xγ^+β^1−xL−β^−γ^=∑β^≤l^∑α^≤Lcβ^L−β^α^−β^xα^1−xL−α^=∑β^≤l^∑α=Lcβ^α^β^Lα^,α0Lβ^xα^1−xα0=∑α^,α0≤L^∑β^≤α^cβ^α^β^L^β^,β0L^α^,α0xα^1−xα0.


## 4. Main Results

The number of Bernstein coefficients ofn-variable polynomial Fx of maximum degree L is equal to D≔L+nL. We aim to store and represent the minimum and maximum Bernstein coefficients in a fast determination method.In the following proposition, we show the linear combination property of Bernstein coefficients.Proposition 3.
ConsiderF be in the Bernstein basis over a given simplex. The coefficients of Bernstein of maximum degree L obtain a linear combination of coefficients of lower degree l.Proof.
Let the Bernstein formF of maximum degree l be given on Δ,(19)Fx=∑α^+α0=lCα^,α0F,l,ΔSα^,α0l.
We deduce from the Bernstein basis that(20)Sα^,α0l=lα^,α0xα^1−xl−α^=lα^,α0xα^1−xα0x+1−xL−l,l≤L=lα^,α0xα^1−xα0∑γ^+γ0=L−lL−lγ^,γ0xγ^1−xγ0=∑γ^+γ0=L−llα^,α0L−lγ^,γ0xα^+γ^1−xα0+γ0α^+γ^≕κ^,α0+γ0≕κ0=∑κ^+τ0=Llα^,α0L−lκ^−α^,κ0−α0xκ^1−xκ0=∑κ^+κ0=Llα^,α0L−lκ^−α^,κ0−α0Lκ^,κ0Lκ^,κ0xκ^1−xκ0=∑κ^+κ0=Llα^,α0L−lκ^−α^,κ0−α0Lκ^,κ0Sκ^,κ0L,from which we have(21)Cκ^,κ0F,L,Δ=∑α^+α0=llα^,α0L−lκ^−α^,κ0−α0Lκ^,κ0Cα^,α0F,l,Δ.Remark 2.
ConsiderFx be a given polynomial of degree L over a simplex. The Bernstein coefficients of ∂F/∂xμx can be calculated by taking linear combinations of Bernstein coefficients of F of degree L−1, i.e., for all α=L−1,(22)CαFi′,L−1,Δ=LCα+ei−Cα.Example 1.
LetFx1,x2≔2x1x22−0.6x12+x1+0.5 be given over the standard simplex Δ. The computed Bernstein coefficients of ∂F/∂x1x are as follows:(23)Cα^F,l,Δ=1130.40.4−0.2.
For determining the enclosure boundary in only one dimensional polynomial of degree 2, we need to compute 6 coefficients.Remark 3.
The minimum value of the enclosure bound optimizes the minimum range of its original polynomial (see Figure2). For positivity analysis of polynomial systems, if the minimum enclosure is positive, then we certify the global positivity of its polynomial over the given domain.Figure 2
The surface of a bi-variate polynomial over the standard simplex, the coefficients of Bernstein marked by black points, and the minimum enclosure bound (blue surface) bounds its polynomial surface over the whole domain.


### 4.1. The Implicit Bernstein Form

In this subsection, we provide a method of computing the implicit Bernstein coefficients overΔ. Let(24)Fx=Fx1+⋯+Fxt,be a given polynomial power form comprising the monomials Fxr,r=1,…,t. Let β^r∈ℕn, r∈1,…,t, then F can be written as(25)Fx=cβ^1xβ^1+⋯+cβ^tcβ^t,withβ^r=l,for some r∈1,…,t.For avoiding the constant terms, we assume thatβ^r>0 for all r=1,…,t.IfF consists t terms of monomials, then each Bernstein coefficient of degree L can be added to the corresponding Bernstein coefficient of the next term:(26)Cα^,α0F,L,Δ=∑r=1tCα^r,α0rrF,L,Δ,α^r+α0r=L,where Cα^r,α0rrF,L,Δ are the coefficients of the r th term of Bernstein F.Remark 4.
ConsiderFx and Px be in Bernstein form of the same degree L. Then, we have(27)Fx+Px=∑α=LCαF,L,Δ+CαP,L,ΔSαx.Example 2.
LetFx1,x2=2x1x2−0.1x13+5 of degree l=3 be given over the standard simplex Δ=e0,e1,e2. Adding Bernstein coefficients to the corresponding coefficients of each term gives the total number of Bernstein coefficients for Fx:(28)Cα^F,l,Δ=0000013230230+000000000−0.1+5555555555=5555516317351734.5.
The minimum and maximum values ofCα^F,l,Δ approximate F, and the number of computed coefficients is D=10.Remark 5.
LetFx be of degree LF and px be of degree Lp with Lp≤LF. Then, we have(29)F+p=∑α=LFCαF,LF,Δ+∑β=minLp,αLpβ.LF−Lpα−βLFαCβp,Lp,ΔSαLF.


### 4.2. Monotonicity of Monomial Coefficients

In this subsection, we show the monotonicity of Bernstein coefficients for high dimensional monomials over a simplex. We assumet=1 and F^x is an n− variable monomial.Lemma 1.
LetF^x=cβ^xβ^, x∈ℝn, be a monomial and e0,…,en denote the canonical basis of ℝn+1. Then, the Bernstein coefficients Cα^,α0F^,L,Δ are monotone with respect to α^, i.e.,(30)Cα^F^,L,Δ≤Cα^+eiF^,L,Δ,ifcβ^>0,Cα^F^,L,Δ≥Cα^+eiF^,L,Δ,ifcβ^<0.Proof.
LetF^x=cβ^xβ^, 0<β^≤l, be a multivariate power form monomial of degree l on Δ. We can express F^x in the monomial Bernstein form of degree L≥l. Assume without loss the generality that cβ^>0. Then, we conclude that(31)Cα^F^,L,Δ=α^β^Lβ^cβ^≤α^+eiβ^Lβ^cβ^=Cα^+eiF^,L,Δ,for. i∈0,…,n.


### 4.3. Polynomials under Dominance

In this subsection, we consider polynomials (of two terms)Fx=Fx++Fx− of orders β^1,β^2, respectively, and cβ^1>0,cβ^2<0. Assume for simplicity that l=L. We provide a direct determination of maxCαF and minCαF that occur at some a^+,a^∗∈ℕn, respectively.Proposition 4.
If∀α^<landγ^=l,γ^∈ℕn,(32)Cα^F−,l,Δ+be0F+,l,Δ>Cα^+eiF−,l,Δ+Cγ^F+,l,Δ,then α^+=e0 and α^∗=α^for someα^ obtains α^=l with α0=0.If∀α^<landγ^=l(33)Cγ^F−,l,Δ+Cα^+eiF+,l,Δ>Ce0F−,l,Δ+Cα^F+,l,Δ,then α^∗=e0 and α^+=α^for someα^ obtains α^=l with α0=0.Proof.
We give the proof of (32) and the proof of (33) is analogous. For all α^<l and γ^=l, we have(34)Cα^+eiF,l,Δ=Cα^+eiF−,l,Δ+Cα^+eiF+,l,Δ≤Cα^+eiF−,l,Δ+Cγ^F+,l,Δ<Cα^F−,l,Δ+Ce0F+,l,Δ≤Cα^F−,l,Δ+Cα^F+,l,Δ=Cα^F,l,Δ,where Cα^F,l,Δ are decreasing with respect to α^ and the proof follows.


### 4.4. Monotonicity of Polynomial Coefficients

The determination of enclosure boundary for multivariate polynomials in Bernstein form requiredD number of coefficients. The minimum Bernstein coefficient of multivariate polynomials approximates the minimum range over the same domain (see Figure 2). We dramatically reduce the search space of coefficients by obtaining the monotonicity of Bernstein.Remark 6.
ConsiderFx and Px be polynomials in Bernstein form of degree LF and LP, respectively. From [13], we have(35)Fx·Px=∑γ=LF+LP∑β=minLF,γLFβLPγ−βLF+LPγ·CβF,LFCγ−βP,LPSγLF+LPx=∑γ=LF+LPCγF·PSγLF+LPx.Example 3.
LetFx1,x2=2x1x2−0.6x12+x22+0.5 and Px1,x2=0.1x12−0.2x1x2+x1+0.3 be given over Δ. The computed Bernstein coefficients of F·P of degree LF+LP=4 are as follows:(36)CγF·P,4,Δ=0.150.150.20.30.450.2750.31670.49170.750.37830.62831.0450.311.015−0.14.
The number of computed coefficients of degree 4 isD=4+22=15.In the following theorem, we generalize the method of fast determination of Bernstein enclosure boundary to the case of monotonicity.Theorem 2.
The multi-indicesα^∗ and α^+ of the minimum and maximum coefficients of a Bernstein polynomial Fx over a simplex with power form coefficients cβ^r>0cβ^r<0analogous for all r=1,…,t satisfy(37)α^∗=minr=1,…,tα^∗r,(38)α^+=maxr=1,…,tα^+r.Proof.
We give the proof of (37), and the proof of (38) is entirely analogous. Assume that cβ^r>0,r=1,…,t. We proceed by contradiction and assume there is some α^∗, 0≤α^∗≤l, from which(39)α^∗<α^∗r,r∈1,…,t.
It follows that(40)α^∗=e0.
Therefore, we deduce thatα^∗r>0,for allr∈1,…,t. Then, the coefficients of Bernstein Cα^rrp,l,Δ are decreasing with respect to α^r and α^∗r=l for all r∈1,…,t. Additionally, the coefficients Cα^F,l,Δ=∑r=1tCα^rrF,l,Δ are decreasing with respect to α^ and so α^∗>0, which is a contradiction of (40).Example 4.
LetF=3x7+x5+6x8 be a polynomial power form given over I=0,1. By application of Theorem 2 to F, we directly find that α^+=8 and α^∗=0. Subsequently, the respective maximum and minimum Bernstein values appear at the corresponding values of α^+ and α^∗, which can be computed using (17).Corollary 1.
LetFx be a (multivariate) power form polynomial of t terms with coefficients cβ^r>0,∀r=1,…,t. Then, for all r∈1,…,t, the multi-indices of minimum and maximum Bernstein coefficients are appeared as α^∗r=e0 and α^+r=α^r,for someα^r≥β^rwithα^r=l. If the coefficients aβ^r<0,∀r=1,…,t, then we have α^+r=e0, and α^∗r=α^r,for someα^r≥β^rwithα^r=l.Corollary 2.
By Lemma1 and Theorem 2, the number of coefficients that are needed to determine the enclosure boundary of n− dimensional polynomials over a simplex does not exceed 1+n1.


## 4.1. The Implicit Bernstein Form

In this subsection, we provide a method of computing the implicit Bernstein coefficients overΔ. Let(24)Fx=Fx1+⋯+Fxt,be a given polynomial power form comprising the monomials Fxr,r=1,…,t. Let β^r∈ℕn, r∈1,…,t, then F can be written as(25)Fx=cβ^1xβ^1+⋯+cβ^tcβ^t,withβ^r=l,for some r∈1,…,t.For avoiding the constant terms, we assume thatβ^r>0 for all r=1,…,t.IfF consists t terms of monomials, then each Bernstein coefficient of degree L can be added to the corresponding Bernstein coefficient of the next term:(26)Cα^,α0F,L,Δ=∑r=1tCα^r,α0rrF,L,Δ,α^r+α0r=L,where Cα^r,α0rrF,L,Δ are the coefficients of the r th term of Bernstein F.Remark 4.
ConsiderFx and Px be in Bernstein form of the same degree L. Then, we have(27)Fx+Px=∑α=LCαF,L,Δ+CαP,L,ΔSαx.Example 2.
LetFx1,x2=2x1x2−0.1x13+5 of degree l=3 be given over the standard simplex Δ=e0,e1,e2. Adding Bernstein coefficients to the corresponding coefficients of each term gives the total number of Bernstein coefficients for Fx:(28)Cα^F,l,Δ=0000013230230+000000000−0.1+5555555555=5555516317351734.5.
The minimum and maximum values ofCα^F,l,Δ approximate F, and the number of computed coefficients is D=10.Remark 5.
LetFx be of degree LF and px be of degree Lp with Lp≤LF. Then, we have(29)F+p=∑α=LFCαF,LF,Δ+∑β=minLp,αLpβ.LF−Lpα−βLFαCβp,Lp,ΔSαLF.


## 4.2. Monotonicity of Monomial Coefficients

In this subsection, we show the monotonicity of Bernstein coefficients for high dimensional monomials over a simplex. We assumet=1 and F^x is an n− variable monomial.Lemma 1.
LetF^x=cβ^xβ^, x∈ℝn, be a monomial and e0,…,en denote the canonical basis of ℝn+1. Then, the Bernstein coefficients Cα^,α0F^,L,Δ are monotone with respect to α^, i.e.,(30)Cα^F^,L,Δ≤Cα^+eiF^,L,Δ,ifcβ^>0,Cα^F^,L,Δ≥Cα^+eiF^,L,Δ,ifcβ^<0.Proof.
LetF^x=cβ^xβ^, 0<β^≤l, be a multivariate power form monomial of degree l on Δ. We can express F^x in the monomial Bernstein form of degree L≥l. Assume without loss the generality that cβ^>0. Then, we conclude that(31)Cα^F^,L,Δ=α^β^Lβ^cβ^≤α^+eiβ^Lβ^cβ^=Cα^+eiF^,L,Δ,for. i∈0,…,n.


## 4.3. Polynomials under Dominance

In this subsection, we consider polynomials (of two terms)Fx=Fx++Fx− of orders β^1,β^2, respectively, and cβ^1>0,cβ^2<0. Assume for simplicity that l=L. We provide a direct determination of maxCαF and minCαF that occur at some a^+,a^∗∈ℕn, respectively.Proposition 4.
If∀α^<landγ^=l,γ^∈ℕn,(32)Cα^F−,l,Δ+be0F+,l,Δ>Cα^+eiF−,l,Δ+Cγ^F+,l,Δ,then α^+=e0 and α^∗=α^for someα^ obtains α^=l with α0=0.If∀α^<landγ^=l(33)Cγ^F−,l,Δ+Cα^+eiF+,l,Δ>Ce0F−,l,Δ+Cα^F+,l,Δ,then α^∗=e0 and α^+=α^for someα^ obtains α^=l with α0=0.Proof.
We give the proof of (32) and the proof of (33) is analogous. For all α^<l and γ^=l, we have(34)Cα^+eiF,l,Δ=Cα^+eiF−,l,Δ+Cα^+eiF+,l,Δ≤Cα^+eiF−,l,Δ+Cγ^F+,l,Δ<Cα^F−,l,Δ+Ce0F+,l,Δ≤Cα^F−,l,Δ+Cα^F+,l,Δ=Cα^F,l,Δ,where Cα^F,l,Δ are decreasing with respect to α^ and the proof follows.


## 4.4. Monotonicity of Polynomial Coefficients

The determination of enclosure boundary for multivariate polynomials in Bernstein form requiredD number of coefficients. The minimum Bernstein coefficient of multivariate polynomials approximates the minimum range over the same domain (see Figure 2). We dramatically reduce the search space of coefficients by obtaining the monotonicity of Bernstein.Remark 6.
ConsiderFx and Px be polynomials in Bernstein form of degree LF and LP, respectively. From [13], we have(35)Fx·Px=∑γ=LF+LP∑β=minLF,γLFβLPγ−βLF+LPγ·CβF,LFCγ−βP,LPSγLF+LPx=∑γ=LF+LPCγF·PSγLF+LPx.Example 3.
LetFx1,x2=2x1x2−0.6x12+x22+0.5 and Px1,x2=0.1x12−0.2x1x2+x1+0.3 be given over Δ. The computed Bernstein coefficients of F·P of degree LF+LP=4 are as follows:(36)CγF·P,4,Δ=0.150.150.20.30.450.2750.31670.49170.750.37830.62831.0450.311.015−0.14.
The number of computed coefficients of degree 4 isD=4+22=15.In the following theorem, we generalize the method of fast determination of Bernstein enclosure boundary to the case of monotonicity.Theorem 2.
The multi-indicesα^∗ and α^+ of the minimum and maximum coefficients of a Bernstein polynomial Fx over a simplex with power form coefficients cβ^r>0cβ^r<0analogous for all r=1,…,t satisfy(37)α^∗=minr=1,…,tα^∗r,(38)α^+=maxr=1,…,tα^+r.Proof.
We give the proof of (37), and the proof of (38) is entirely analogous. Assume that cβ^r>0,r=1,…,t. We proceed by contradiction and assume there is some α^∗, 0≤α^∗≤l, from which(39)α^∗<α^∗r,r∈1,…,t.
It follows that(40)α^∗=e0.
Therefore, we deduce thatα^∗r>0,for allr∈1,…,t. Then, the coefficients of Bernstein Cα^rrp,l,Δ are decreasing with respect to α^r and α^∗r=l for all r∈1,…,t. Additionally, the coefficients Cα^F,l,Δ=∑r=1tCα^rrF,l,Δ are decreasing with respect to α^ and so α^∗>0, which is a contradiction of (40).Example 4.
LetF=3x7+x5+6x8 be a polynomial power form given over I=0,1. By application of Theorem 2 to F, we directly find that α^+=8 and α^∗=0. Subsequently, the respective maximum and minimum Bernstein values appear at the corresponding values of α^+ and α^∗, which can be computed using (17).Corollary 1.
LetFx be a (multivariate) power form polynomial of t terms with coefficients cβ^r>0,∀r=1,…,t. Then, for all r∈1,…,t, the multi-indices of minimum and maximum Bernstein coefficients are appeared as α^∗r=e0 and α^+r=α^r,for someα^r≥β^rwithα^r=l. If the coefficients aβ^r<0,∀r=1,…,t, then we have α^+r=e0, and α^∗r=α^r,for someα^r≥β^rwithα^r=l.Corollary 2.
By Lemma1 and Theorem 2, the number of coefficients that are needed to determine the enclosure boundary of n− dimensional polynomials over a simplex does not exceed 1+n1.


## 5. Conclusions

In this work, we considered computing the minimum and maximum Bernstein coefficients (enclosure boundary) that optimize the range of polynomials over a simplex. We reduced the high complexity of computing alln-dimensional Bernstein coefficients of degree L to 1+n1. Therefore, we covered the cases of monotonicity, dominance, and multivariate monotonicity of Bernstein coefficients. Finally, the index of enclosure boundary was directly determined and only the enclosure value computed. In the future work, we consider generalizing this method to new classes of functions such as rational polynomial functions over triangles.



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