Approximating amoebas and coamoebas by sums of squares
aa r X i v : . [ m a t h . AG ] J un APPROXIMATING AMOEBAS AND COAMOEBASBY SUMS OF SQUARES
THORSTEN THEOBALD AND TIMO DE WOLFF
Abstract.
Amoebas and coamoebas are the logarithmic images of algebraic varietiesand the images of algebraic varieties under the arg-map, respectively. We present newtechniques for computational problems on amoebas and coamoebas, thus establishingnew connections between (co-)amoebas, semialgebraic and convex algebraic geometryand semidefinite programming.Our approach is based on formulating the membership problem in amoebas (respec-tively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstel-lensatz, this allows to tackle the problem by sums of squares techniques and semidefiniteprogramming. Our method yields polynomial identities as certificates of non-containmentof a point in an amoeba or coamoeba. As the main theoretical result, we establish somedegree bounds on the polynomial certificates. Moreover, we provide some actual compu-tations of amoebas based on the sums of squares approach. Introduction
For an ideal I ⊂ C [ Z , . . . , Z n ] an amoeba (introduced by Gel ′ fand, Kapranov, andZelevinsky [3], see also the surveys [9] or [17]) is the image of the variety V ( I ) whereeach complex coordinate is mapped to (the logarithm of) its absolute value. It is oftencustomary and useful to consider the logarithmic version of an amoeba A I = { (log | z | , . . . , log | z n | ) : z ∈ V ( I ) ∩ ( C ∗ ) n } with C ∗ := C \ { } , and we denote the unlog amoeba by U I = { ( | z | , . . . , | z n | ) : z ∈ V ( I ) } . Similarly, the coamoeba C I is defined as C I := Arg( V ( I ) ∩ ( C ∗ ) n ), where Arg denotes themapping ( z , . . . , z n ) (arg( z ) , . . . , arg( z n )) ∈ T n := ( R / π Z ) n and arg denotes the argument of a complex number (see [11, 12, 13, 14]). If I is a principalideal generated by a polynomial f , we shortly write A f := A h f i and analogously U f , C f .Studying computational questions of amoebas has been initiated in [26], where in par-ticular certain special classes of amoebas (e.g., two-dimensional amoebas, amoebas of Mathematics Subject Classification.
Key words and phrases.
Amoebas, sums of squares, real Nullstellensatz, coamoebas, semidefiniteprogramming.Research supported by DFG grant TH 1333/2-1 and (first author) by the A. v. Humboldt-Foundation.
Grassmannians) were studied. One of the natural and fundamental computational ques-tions is the membership problem, which asks for a given point ( λ , . . . , λ n ) whether thispoint is contained in an (unlog) amoeba respectively coamoeba.In [20] Purbhoo provided a characterization for the points in the complement of ahypersurface amoeba which can be used to numerically approximate the amoeba. His lopsidedness criterion provides an inequality-based certificate for non-containment of apoint in an amoeba, but does not provide an algebraic certificate (in the sense of apolynomial identity certifying the non-containment). The certificates are given by iteratedresultants. With this technique the amoeba can be approximated by a limit process.The computational efforts of computing the resultants are growing quite fast, and theconvergence is slow.A different approach to tackle computational problems on amoebas is to apply suitableNullstellen- or Positivstellens¨atze from real algebraic geometry or complex geometry. Forsome natural problems a direct approach via the Nullstellensatz (applied on a realificationof the problem) is possible. Using a degree truncation approach, this allows to find sum-of-squares-based polynomial identities which certify that a certain point is located outsideof an amoeba or coamoeba. In particular, it is well known from recent lines of researchin computational semialgebraic geometry (see, e.g., [4, 5, 15]) that these certificates canbe computed via semidefinite programming (SDP).In this paper, we discuss theoretical foundations as well as some practical issues ofsuch an approach, thus establishing new connections between amoebas, semialgebraicand convex algebraic geometry and semidefinite programming. Firstly, we present variousNullstellensatz-type formulations (a standard approach in Statement 3.2 and a monomialapproach in Statement 3.7) and compare their properties to a recent toric Nullstellensatzof Niculescu and Putinar [10]. Using a degree truncation approach this yields a sequenceof supersets of the amoeba A f , which converges to A f (Theorem 3.11). For every fixedsuperset in this sequence, the membership problem can be solved by semidefinite pro-gramming.The main theoretical contribution is contained in Section 4. For one of our approaches,we can provide some degree bounds for the certificates (Corollary 4.5). It is remark-able and even somewhat surprising that these degree bounds are derived from Purbhoo’slopsidedness criterion (which is not at all sum-of-squares-based). We also show that incertain cases (such as for the Grassmannian of lines) the degree bounds can be reducedto simpler amoebas (Theorem 4.7).In Section 5 we provide some actual computations on this symbolic-numerical approach.Besides providing some results on the membership problem itself, we will also considermore sophisticated versions (such as bounding the diameter of a complement componentfor certain classes).Finally, in Section 6 we give an outlook on further questions on the approach initiatedin the current paper. PPROXIMATING AMOEBAS AND COAMOEBAS 3
Figure 1.
The amoeba of f = Z Z + Z Z − Z Z + 1.2. Preliminaries
In the following, let C [ Z ] = C [ Z , . . . , Z n ] denote the polynomial ring over C in n variables. For f = P α ∈ A c α Z α ∈ C [ Z ], the Newton polytope New( f ) = conv { α ∈ N n : α ∈ supp( f ) } of f is the convex hull of the exponent vectors, where supp( f ) denotes thesupport of f .2.1. Amoebas and coamoebas.
We recall some basic statements about amoebas (see[2, 9, 17]). For any ideal I ⊂ C [ Z ], the amoeba A I is a closed set. For a polynomial f ∈ C [ Z ], the complement of the hypersurface amoeba A f consists of finitely manyconvex regions, and these regions are in bijective correspondence with the different Laurentexpansions of the rational function 1 /f . See Figure 1 for an example.The order ν of a point w in the complement R n \ A f is given by ν j = 1(2 πi ) n Z Log − ( w ) z j ∂ j f ( z ) f ( z ) dz · · · dz n z · · · z n , ≤ j ≤ n , where Log( z ) is defined as Log( z ) = (log | z | , . . . , log | z n | ). The order mapping induces aninjective map from the set of complement components into New( f ) ∩ Z n . The complementcomponents corresponding to the vertices of New( f ) do always exist [2].Similarly, for f ∈ C [ Z ] any connected component of the coamoeba complement T n \ C f is a convex set (see Figure 2). If C f denotes the closure of C f in the torus T n , then thenumber of connected components of T n \ C f is bounded by n ! vol New( f ) (Nisse [14, Thm.5.19]), where vol denotes the volume.For technical reasons (see Theorem 3.6) it will be often convenient to consider in thedefinition of a coamoeba also those points z ∈ V ( I ) which have a zero-component. Namely,if a zero z of I has a zero-component z j = 0 then we associate this component to any phase. Call this modified version of a coamoeba C ′ I , i.e., C ′ I := { φ ∈ T n : ∃ z ∈ V ( I ) : arg( z j ) = φ j or z j = 0 for 1 ≤ j ≤ n } . THORSTEN THEOBALD AND TIMO DE WOLFF
Figure 2.
The coamoeba of f = Z Z + Z Z − Z Z + 1 in two differentviews of T , namely [ − π, π ) versus [0 , π ) .Note that for principal ideals I = h f i the difference between C I and C ′ I solely may occurat points which are contained in the closure of C I . The set-theoretic difference of C I and C ′ I is a lower-dimensional subset of R n (since in each environment of a point in C ′ I \ C I wehave a coamoeba point).2.2. The situation at ∞ . It is well-known that the geometry of amoebas at infinity(i.e., the “tentacles”) can be characterized in terms of logarithmic limit sets and tropicalgeometry, and thus amoebas form one of the building blocks of tropical geometry (forgeneral background on tropical geometry we refer to [8, 9, 22]).For (large)
R > A ( R ) denote the scaled version A ( R ) I := R A I ∩ S n − , where S n − denotes the ( n − logarithmiclimit set A ( ∞ ) I is the set of points v ∈ S n − such that there exists a sequence v R ∈ A ( R ) I with lim R →∞ v R = v . For a polynomial f = P α c α Z α ∈ C [ Z ] denote by trop f := L ⊙ Z α ⊙ · · · ⊙ Z α n n its tropicalization (with respect to the trivial valuation) over thetropical semiring ( R, ⊕ , ⊙ ) := ( R ∪ {−∞} , max , +). Then (see [8, 24]): Proposition 2.1.
A vector w ∈ R n \ { } is contained in the tropical variety of I if andonly if the corresponding unit vector || w || w is contained in A ( ∞ ) I . Thus the tropical varietyof I coincides with the cone over the logarithmic limit set A ( ∞ ) I . Approximations based on the real Nullstellensatz
We study certificates of points in the complement of the amoeba based on the realNullstellensatz and compare them to existing statements in the literature (such as thetoric Nullstellensatz of Niculescu and Putinar). By imposing degree truncations this willthen yield a hierarchy of certificates of bounded degree.We use the following real Nullstellensatz (see, e.g., [1, 19]):
Proposition 3.1.
For polynomials g , . . . , g r ∈ R [ X ] and I := h g , . . . , g r i ⊂ R [ X ] thefollowing statements are equivalent: PPROXIMATING AMOEBAS AND COAMOEBAS 5 • The real variety V R ( I ) is empty. • There exist a polynomial G ∈ I and a sum of squares polynomial H with G + H + 1 = 0 . Given λ ∈ (0 , ∞ ) n , the question if λ is contained in the unlog amoeba U I can bephrased as the real solvability of a real system of polynomial equations. For a polynomial f ∈ C [ Z ] = C [ Z , . . . , Z n ] let f re , f im ∈ R [ X, Y ] = R [ X , . . . , X n , Y , . . . , Y n ] be its realand imaginary parts, i.e., f ( Z ) = f ( X + iY ) = f re ( X, Y ) + i · f im ( X, Y ) . We consider the ideal I ′ ⊂ R [ X, Y ] generated by the polynomials(3.1) { f re j , f im j : 1 ≤ j ≤ r } ∪ (cid:8) X k + Y k − λ k : 1 ≤ k ≤ n (cid:9) . Corollary 3.2.
Let I = h f , . . . , f r i , and λ ∈ (0 , ∞ ) n . Either the point λ is containedin U I , or there exist a polynomial G ∈ I ′ ⊂ R [ X, Y ] and a sum of squares polynomial H ∈ R [ X, Y ] with (3.2) G + H + 1 = 0 . Proof.
For any polynomial f ∈ C [ Z ] it suffices to observe that a point z = x + iy iscontained in V ( f ) if and only if ( x, y ) ∈ V R ( f re ) ∩ V R ( f im ), and that | z k | = λ k if and only( x, y ) ∈ V R ( X + Y − λ k ). Then the statement follows from Proposition 3.1. (cid:3) Corollary 3.2 states that for any point λ
6∈ U I there exists a certificate r X j =1 p j f re j + r X j =1 p ′ j f im j + n X k =1 q k ( X k + Y k − λ k ) + H + 1 = 0(3.3)with polynomials p j , p ′ j , q k and a sum of squares H . We refer to these certificates as certificates in the standard approach .We say that a certificate of the form (3.3) is of degree at most d if the (total) degree ofeach summand in (3.3) is at most d . Remark 3.3.
By the following fact (which is easy to check), the sum of squares condi-tion 3.2 can also be stated shortly as − R [ X, Y ] /I ′ . Fact 3.4. (Parrilo [16].)
Let I = h g , . . . , g r i ⊂ R [ X ] and f ∈ R [ X ] . There exist p , . . . , p r ∈ R [ X ] such that f + X i p i g i is a sum of squares in R [ X ] if and only if f is a sum of squares in R [ X ] /I . THORSTEN THEOBALD AND TIMO DE WOLFF
Any two of these equivalent conditions in Fact 3.4 is a certificate for the nonnegativityof f on the variety I .Before stating a coamoeba version, we note the following normalization properties.Whenever it is needed for amoebas, we can assume that the point λ in the amoebamembership problem is the all-1-vector ∈ R n . Similarly, for the coamoeba membershipproblem we can assume that the investigated point is the origin 0 ∈ T n . Lemma 3.5.
Let I = h f , . . . , f r i . (1) A point ( λ , . . . , λ n ) ∈ (0 , ∞ ) n is contained in U I if and only if ∈ U h g ,...,g r i , where g j ( Z , . . . , Z n ) := f j ( λ Z , . . . , λ n Z n ) , ≤ j ≤ r . (2) A point ( z , . . . , z n ) is contained in V ( I ) with arg z j = µ j if and only if the (non-negative) real vector y with y j := z j e − iµ j is contained in V ( g , . . . , g r ) where g j ( Z , . . . , Z n ) := f j ( Z e iµ , . . . , Z n e iµ n ) , ≤ j ≤ r . Proof.
A point ( z , . . . , z n ) is contained in V ( I ) with | z j | = λ j if and only if the vector y defined by y j := z j /λ j is contained in V ( g , . . . , g r ) with | y j | = 1. The second statementfollows similarly. (cid:3) Theorem 3.6.
Let I = h f , . . . , f r i . The point (0 , . . . , is contained in the complementof the coamoeba C ′ I if and only if there exists a polynomial identity (3.4) r X j =1 c j · f j ( X , Y ) re + r X j =1 c ′ j · f j ( X , Y ) im + n X k =1 d k · Y k + H + 1 = 0 with polynomials c j , c ′ j , d k ∈ R [ X, Y ] and a sum of squares H . Here, f j ( X , Y ) abbreviates f j ( X , . . . , X n , Y , . . . , Y n ) .Proof. Note that the statement 0 ∈ C ′ I is equivalent to { z = x + iy ∈ C n : z ∈V ( I ) and x k ≥ , y k = 0 , ≤ k ≤ n } 6 = ∅ . Moreover, observe that the condition x k ≥ X k in the arguments of f , . . . , f r . Hence, by Proposi-tion 3.1 the statement 0
6∈ C ′ I is equivalent to the existence of a polynomial identity of theform (3.4). (cid:3) Observe that in the proof the use of C ′ I (rather than C I ) allowed to use the basicNullstellensatz (rather than a Positivstellensatz, which would have introduced severalsum of squares polynomials).The following variant of the Nullstellensatz approach will allow to obtain degree bounds(see Section 4). For vectors α (1) , . . . , α ( d ) ∈ N n and coefficients b , . . . , b d ∈ C ∗ let f = P dj =1 b j · Z α ( j ) ∈ C [ Z ]. For any given values of λ , . . . , λ n set µ j := λ α ( j ) = λ α ( j ) · · · λ α ( j ) n n , ≤ j ≤ d . If the rank of the matrix with columns α (1) , . . . , α ( d ) is n (i.e., the vectors α (1) , . . . , α ( d )span R n ) then the λ -values can be reconstructed uniquely from the µ -values. We comeup with the following variant of a Nullstellensatz. PPROXIMATING AMOEBAS AND COAMOEBAS 7
Here, let I := h f , . . . , f r i such that f i is of the form P d i j =1 b ij Z α ( i,j ) with α ( i, j ) ∈ N n .Let m ij be the monomial m ij = Z α ( i,j ) = Z α ( i,j ) · · · Z α ( i,j ) n n . We consider the ideal I ∗ ⊂ R [ X, Y ] generated by the polynomials(3.5) { f re i , f im i : 1 ≤ i ≤ r } ∪ (cid:8) ( m re ij ) + ( m im ij ) − µ ij : 1 ≤ j ≤ d i , ≤ i ≤ r (cid:9) , where µ ij = λ α ( i,j ) . Corollary 3.7.
Let I := h f , . . . , f r i , and assume that the set S ri =1 S d i j =1 { α ( i, j ) } spans R n . Either a point λ ∈ (0 , ∞ ) n is contained in U I , or there exist polynomials G ∈ I ∗ ⊂ R [ X, Y ] and a sum of squares polynomial H ∈ R [ X, Y ] with (3.6) G + H + 1 = 0 . We refer to these certificates as certificates in the monomial approach .For hypersurface amoebas of real polynomials, the membership problem relates to thefollowing statement of Niculescu and Putinar [10]. Let p = p ( X, Y ) ∈ R [ X , . . . , X n ,Y , . . . , Y n ] be a real polynomial. Then p can be written as a complex polynomial p ( X, Y ) = P ( Z, Z ) with P ∈ C [ Z . . . , Z n , ¯ Z , . . . , ¯ Z n ] and P ( Z, Z ) = P ( Z, ¯ Z ). Note that there existsa polynomial Q ∈ C [ Z , . . . , Z n ] with p ( x, y ) = | P ( z, ¯ z ) | = | Q ( z ) | for z ∈ T n , where T := { z ∈ C : | z | = 1 } .The following statement can be obtained by applying the Nullstellensatz on the set { z = ( x, y ) : | q ( z ) | = 1 , | z | = 1 , . . . , | z n | = 1 } , then applying Putinar’s Theorem [21]on the multiplier polynomial of | q ( Z ) | (see [10]). Proposition 3.8.
Let q ∈ C [ Z , . . . , Z n ] . Then q ( z ) = 0 for all z ∈ T n if and only ifthere are complex polynomials p , . . . , p k , r , . . . , r l ∈ C [ Z , . . . , Z n ] with (3.7) 1 + | p ( z ) | + · · · + | p k ( z ) | = | q ( z ) | ( | r ( z ) | + · · · + | r l ( z ) | ) , z ∈ T n . Note that the statement is not an identity of polynomials, but an identity for all z inthe n - torus T n .While Proposition 3.8 provides a nice structural result, due to the following reasonswe prefer Corollary 3.2 for actual computations. In representation (3.7), two sums ofsquares polynomials (rather than just one as in (3.2)) are needed in the representation,and the degree is increased (by the squaring process). Moreover, the theorem is not reallya representation theorem (in terms of an identity of polynomials), but an identity over T n ; therefore in order to express this computationally, the polynomials hidden in thisequivalence (i.e., the polynomials 1 − | Z | , . . . , − | Z n | ) have to be additionally used. SOS-based approximations.
By putting degree truncations on the certificates, we cantransform the theoretic statements into effective algorithmic procedures for constructingcertificates. The idea of degree truncations in polynomial identities follows the same prin-ciples of the degree truncations with various types of Nullstellen- and Positivstellens¨atzein [4, 6, 15]. It is instructive to have a look at two simple examples first.
THORSTEN THEOBALD AND TIMO DE WOLFF
Example 3.9.
Let f be the polynomial f = Z + z with a complex constant z = x + iy .The ideal I ′ of interest is defined by h := f re = X + x ,h := f im = Y + y ,h := X + Y − λ . For values of λ ≥ λ = x + y ), wehave V C ( I ) = ∅ and thus the Gr¨obner basis G of h h , h , h i is G = { } . The correspondingmultiplier polynomials p j to represent 1 as a linear combination P j p j h j are p = − X + x x + y − λ , p = − Y + y x + y − λ , p = 1 x + y − λ . Hence, in particular, − R [ X ] /I ′ .The necessary degree with regard to equation (3.2) is just 2.For λ = x + y , the Gr¨obner basis (w.r.t. a lexicographic variable ordering with X ≻ Y ) is X + x , Y + y . The point ( − x , − y ) is contained in V R ( I ′ ); thus in this case there does not exist aNullstellen-type certificate. Example 3.10.
Consider the polynomial f = Z + Z + 5 with Z j = X j + iY j . The ideal I ′ of interest is defined by h := X + X + 5 , h := Y + Y , h := X + Y − λ , h := X + Y − λ . Consider λ = 2, λ = 3. Using a lexicographic ordering with X ≻ X ≻ Y ≻ Y , aGr¨obner basis is Y , Y + Y , X + 3 , X + 2 . The standard monomials are 1 and Y . It is easy to see that − , ∈ U I .Consider now the choice λ = 1 and λ = 2. Using lexicographic ordering again, theGr¨obner basis is 25 Y + 96 , Y + Y , X + 14 , X + 11 . Hence, 25 Y ≡ −
96 mod I ′ , which gives the sum of squares identity (cid:18) √ Y (cid:19) ≡ − I ′ , and thus shows (1 , / ∈ U I .Using the degree truncation approach for sums of squares we can for a given ideal I = h f , . . . , f r i define C t := { λ ∈ (0 , ∞ ) n \ U I : there exists a certificate of the form (3.3)for λ of degree ≤ t } . PPROXIMATING AMOEBAS AND COAMOEBAS 9
Similarly, for coamoebas C I let D t be the subsets of its complement obtained by thedegree truncation. These sequences are the basis of the effective implementation (seeSection 5). Theorem 3.11.
Let I = h f , . . . , f r i and t := max j ⌈ deg f j / ⌉ . The sequence ( C t ) t ≥ t converges pointwise to the complement of the unlog amoeba U I , and it is monotone in-creasing in the set-theoretic sense, i.e. C t ⊂ C t +1 for t ≥ t .Similarly, the sequence ( D t ) t ≥ t converges pointwise to the complement of the coamoeba C I , and it is monotone increasing in the set-theoretic sense, i.e. D t ⊂ D t +1 for t ≥ t .Proof. For any given point z in the complement of the amoeba there exists a certificate ofminimal degree, say d . For t < ⌈ d/ ⌉ the point z is not contained in C t and for t ≥ ⌈ d/ ⌉ the point z is contained in C t . In particular, the relaxation process is monotone increasing.And analogously for coamoebas. (cid:3) Remark 3.12.
A similar result holds for the monomial approach. Namely, for a givenideal I = h f , . . . , f r i , t := max j ⌈ deg f j / ⌉ , and I ∗ generated by the polynomials (3.5),the sets C ∗ t := { λ ∈ (0 , ∞ ) n \ U I : there exists a certificate of the form G + H + 1 = 0with G ∈ I ∗ and H SOS for λ of degree ≤ t } ( t ≥ t ) converge pointwise to the complement of U I , and, set-theoretically, this sequenceis monotone increasing.It is well-known (and at the heart of current developments in optimization of polynomialfunctions, see [4, 15] or the survey [5]) that SOS conditions with degree constraints of theform (3.3) can be phrased as semidefinite programs. Finding an (optimal) positive semi-definite matrix within an affine linear variety is known as semidefinite programming (seee.g. [28] for a comprehensive treatment). Semidefinite programs can be solved efficientlyboth in theory and in practice.Precisely, any sum-of-squares polynomial H can be expressed as M QM T , where Q is a symmetric positive semidefinite matrix (abbreviated Q (cid:23)
0) and M is a vector ofmonomials.Similarly, by the degree restriction the linear combination in (3.2) or (3.6) can beintegrated into the semidefinite formulation by a comparison of coefficients.4. Special certificates
For a certain class of amoebas, we can provide some explicit classes of Nullstellensatz-type certificates. As a first warmup-example, we illustrate some ideas for constructingspecial certificates systematically for linear amoebas in the standard approach. Then weshow how to construct special certificates for the monomial-based approach.In this section we concentrate on the case of hypersurface amoebas.
Linear amoebas in the standard approach.
Let f = aZ + bZ + c be a general linearpolynomial in two variables with real coefficients a, b, c ∈ R . We consider certificates of the form (3.2) based on the third binomial formula ( α + β )( α − β ) = α − β , and we usethe sums of squares ( X − X ) and ( Y − Y ) . For simplicity assume a, b >
0. Setting G := ( aX + bX − c )( aX + bX + c ) + ( aY + bY )( aY + bY ) − ( a + ab )( X + Y − λ ) − ( b + ab )( X + Y − λ ) ,H := ab ( X − X ) + ab ( Y − Y ) the sum G + H simplifies via elementary cancellation to γ := ( a + ab ) λ + ( b + ab ) λ − c . (4.1)Assume that the point ( λ , λ ) is not contained in the unlog amoeba U f . In order toobtain the desired polynomial identity (3.2) certifying containment in the complement of U f , we require γ to be negative. In that case we have1 | γ | ( G + H ) + 1 = 0 , which gives the polynomial identity (3.2). Analogously, we obtain for G := ( − aX + bX + c )( aX + bX + c ) + ( − aY + bY )( aY + bY ) a ( X + Y − λ ) − ( b + bc )( X + Y − λ ) ,H := bc ( X − + bcY via elementary cancellation γ := G + H = ( b + bc ) λ + c + bc − a λ , and, symmetrically, γ := ( a + ac ) λ + c + ac − b λ . Example 4.1.
Let a = 1, b = 2, c = 5. The curve (in λ , λ ) given by (4.1) hasa logarithmic image that is shown in Figure 3. Analogous special certificates can beobtained within the two other complement components. –2–112–3 –2 –1 1 2 3 4 Figure 3.
The boundary of an amoeba of a linear polynomial (blue) andthe logarithmic image of the boundary of R = { λ ∈ R : ( a + ab ) λ +( b + ab ) λ − c < } , for which the special certificates of degree 2 exist(red). PPROXIMATING AMOEBAS AND COAMOEBAS 11
Since (by homogenizing the polynomial) there is a symmetry, we obtain similarly anapproximation of the other two complement components. Hence we have:
Lemma 4.2.
Let f = aZ + bZ + c with a, b, c ∈ R , and set R = { λ ∈ R : ( a + ab ) λ + ( b + ab ) λ − c < } ,R = { λ ∈ R : ( b + bc ) λ + c + bc − a λ < } ,R = { λ ∈ R : ( a + ac ) λ + c + ac − b λ < } . Then R j ∩ U f = ∅ for ≤ j ≤ , and for any λ ∈ R ∪ R ∪ R there exists a certificate (3.3) of degree at most two. The monomial-based approach.
For the monomial-based approach based on Corol-lary 3.7 we can provide special certificates for a much more general class. Our point ofdeparture is Purbhoo’s lopsidedness criterion [20] which guarantees that a point belongsto the complement of an amoeba A f . In particular, we can provide degree bounds forthese certificates.In the following let α (1) , . . . , α ( d ) ∈ N n span R n and f = P dj =1 b j Z α ( j ) ∈ C [ Z ] withmonomials m j := Z α ( j ) = Z α ( j ) · · · Z α ( j ) n n . For any point v ∈ R n define f { v } as thefollowing sequence of numbers in R > , f { v } := (cid:0) | b m (Log − ( v )) | , . . . , | b d m d (Log − ( v )) | (cid:1) . A list of positive real numbers is called lopsided if one of the numbers is greater than thesum of all the others. We call a point v ∈ R n lopsided , if the sequence f { v } is lopsided.Furthermore set LA ( f ) := { v ∈ R n : f { v } is not lopsided } . It is easy to see that A f ⊂ LA ( f ) with A f = LA ( f ) in general. In the following way theamoeba A f can be approximated based on the lopsidedness concept. For r ≥ f r ( Z ) := r − Y k =0 · · · r − Y k n =0 f (cid:0) e πik /r Z , . . . , e πik n /r Z n (cid:1) = Res (cid:0) Res (cid:0) . . .
Res( f ( U Z , . . . , U n Z n ) , U r − , . . . , U rn − − (cid:1) , U rn − (cid:1) , where Res denotes the resultant of two (univariate) polynomials. Then the followingtheorem holds. Theorem 4.3. (Purbhoo [20, Theorem 1])(a)
For r → ∞ the family LA ( ˜ f r ) converges uniformly to A f . There exists an integer N such that to compute A f within ε > , it suffices to compute LA ( ˜ f r ) for any r ≥ N . Moreover, N depends only on ε and the Newton polytope (or degree) of f and can be computed explicitly from these data. (b) For an ideal I ⊂ C [ Z ] a point v ∈ R n is in the amoeba A I if and only if g { v } isnot lopsided for every g ∈ I . Based on Theorem 4.3 one can devise a converging sequence of approximations forthe amoeba. Note, however, that the lopsidedness criterion is not a Nullstellensatz in astrict sense since it does not provide a polynomial identity certifying membership in thecomplement of the amoeba.The aim of this section is to figure out how our SOS approximation is related to thelopsidedness and transform the lopsidedness certificate into a certificate for the Nullstel-lens¨atze presented in Section 3.By Lemma 3.5 we can assume that the point λ , whose membership to an unlog amoeba U f shall be decided, is the all-1-vector . In this situation lopsidedness means that thereis an index j ∈ { , . . . , d } with | b j | > P i = j | b i | . If the lopsidedness condition is satisfied in v , then the following statement provides a certificate of the form G + H + 1 and boundeddegree.Corresponding to the definition of I ∗ in (3.5), let the polynomials s , . . . , s d +2 be definedby s i = (cid:18) b re i | b i | · (cid:0) Z α ( i ) (cid:1) re (cid:19) + (cid:18) b im i | b i | · (cid:0) Z α ( i ) (cid:1) im (cid:19) − , ≤ i ≤ d , and s d +1 = f re , s d +2 = f im . Theorem 4.4.
If the point λ = is contained in the complement of U f with f { } beinglopsided with dominating element | m ( ) | , then there exists a certificate of (total) degree · deg( f ) which is given by (4.2) d +2 X i =1 s i g i + H + 1 = 0 , where g = | b | , g i = −| b i | · d X k =2 | b k | , ≤ i ≤ d ,g d +1 = − b · Z α (1) + d X i =2 b i · Z α ( i ) ! re , g d +2 = − b · Z α (1) + d X i =2 b i · Z α ( i ) ! im ,H = X ≤ i By the third binomial formula ( α + β ) · ( α − β ) = α − β , substituting thepolynomials s i and g j into s d +1 g d +1 + s d +2 g d +2 yields − (cid:16) b re1 · (cid:0) Z α (1) (cid:1) re (cid:17) + d X i =2 ( b i · Z α ( i ) ) re ! − (cid:16) b im1 · (cid:0) Z α (1) (cid:1) im (cid:17) + d X i =2 ( b i · Z α ( i ) ) im ! . PPROXIMATING AMOEBAS AND COAMOEBAS 13 Adding g s and the SOS term H yields −| b | + d X j =2 (cid:18) b re j | b j | · (cid:0) Z α ( j ) (cid:1) re (cid:19) + (cid:18) b im j | b j | · (cid:0) Z α ( j ) (cid:1) im (cid:19) ! · | b j | · d X k =2 | b k | ! . Hence, the expression P d +2 i =1 s i g i + H in (4.2) in total results in −| b | + d X i =2 | b i | ! . Since all | b i | ≥ | m ( ) | ,this is the certificate we wanted to obtain. By rescaling, we can bring the constant to − (cid:3) We say that there exists a certificate for a point w in the complement of the (log)amoeba A f if there exists a certificate for the point in the complement of the amoeba U g in the sense of Theorem 4.4, where g is defined as in Lemma 3.5 and λ i := | log − ( w i ) | . Corollary 4.5. Let r ∈ N . (1) For any w ∈ R n \ LA ( ˜ f r ) ⊂ R n \ A f there exists a certificate of degree at most · r n · deg( f ) which can be computed explicitly. (2) The certificate determines the order of the complement component to which w belongs.Proof. By definition of g , we have w ∈ A f if and only if ∈ U g . Further belongs to LA (˜ g r ) if and only if ˜ g r { } is not lopsided. Applying Theorem 4.4 on the function ˜ g r yieldsa certificate for w in the log amoeba A f . Since we have tdeg(˜ g r ) = tdeg( g ) · r n = tdeg( f ) · r n due to the definition of ˜ g r and g the result follows.For the second statement, note that passing over from f to g does not change theorder ν of any point in the complement of the amoebas. Now it suffices to show that thedominating term (which occurs in a distinguished way in the certificate) determines theorder of the complement component. The latter statement follows from Purbhoo’s resultthat if w 6∈ LA ( ˆ f r ) and the order of the complement component w belongs to is α ( i ) thenthe dominant term in ˜ f r has the exponent r n · α ( i ) (see [20, Proposition 4.1]). (cid:3) Corollary 4.6. For linear hyperplane amoebas in R n , any point in the complement of theamoeba has a certificate whose sum of squares is a sum of squares of affine functions.Proof. By the explicit characterization of linear hyperplane amoebas in [2, Corollary 4.3],any point in the complement is lopsided. Hence, the statement follows from Theorem 4.4. (cid:3) Simplified expressions. From a slightly more general point of view, the monomial-based certificates can be seen as a special case of the following construction. Wheneverthe defining polynomials of a variety originate from simpler polynomials with algebraicallyindependent monomials, then the approximation of the amoeba can be simplified. For an ideal I let V := V ( I ) ⊂ ( C ∗ ) n be its subvariety in ( C ∗ ) n . Let γ , . . . , γ k be k monomials in n variables, say, γ i = Y α ( i ) = Y α ( i ) Y α ( i ) · · · Y α ( i ) n n , where α ( i ) =( α ( i ) , . . . , α ( i ) n ) ∈ Z n . They define a homomorphism γ of algebraic groups from ( C ∗ ) n to ( C ∗ ) k . For any subvariety W of ( C ∗ ) k , the inverse image γ − ( W ) is a subvariety of( C ∗ ) n . Note that the map γ is onto if and only if the vectors α (1) , . . . , α ( k ) are linearlyindependent (see [26, Lemma 4.1]).Let J be an ideal with V ( J ) = γ − ( V ). If the map γ is onto, then computing theamoeba of J can be reduced to the computation of the amoeba of I . Let γ ′ denote therestriction of γ to the multiplicative subgroup (0 , ∞ ) n . Then the following diagram is acommutative diagram of multiplicative abelian groups:( C ∗ ) n γ −→ ( C ∗ ) k ↓ ↓ (0 , ∞ ) n γ ′ −→ (0 , ∞ ) k where the vertical maps are taking coordinate-wise absolute value. For vectors p =( p , . . . , p n ) in ( C ∗ ) n we write | p | = ( | p | , . . . , | p n | ) ∈ (0 , ∞ ) n , and similarly for vectors oflength k . Further, for V ⊂ ( C ∗ ) n let | V | := {| p | : p ∈ V } . If the map γ is onto then | γ − ( V ) | = γ ′− ( | V | ) (see [26]). Theorem 4.7. If a point outside of an unlog amoeba U I has a certificate of total degree d then a point outside of the unlog U J has a certificate of degree d · D , where D is themaximal total degree of the monomials γ , . . . , γ k . In particular, this statement applies to the certificates from Statements 3.2 and 3.7. Proof. Let p be a point outside of the unlog amoeba of V ⊂ ( C ∗ ) n which has a certificateof total degree d . By Corollary (3.2), the certificate consists of a polynomial G ( X, Y )in the real ideal I ′ ⊂ R [ X, Y ] from (3.1) and by real sums of squares of polynomials in R [ X, Y ]. For the polynomials in the ideals, we observe that the realification process carriesover to the substitution process. W.l.o.g. we can assume that γ i is a product of just twofactors. Then, with Z = Z + iZ , Z = P Q we have Z + iZ = ( P + iP )( Q + iQ ) anduse the real substitutions Z ≡ P Q − P Q , Z ≡ P Q + P Q . And in the same waythe real sum of squares remain real sums of squares (of the polynomials in P i , Q j ) aftersubstituting. (cid:3) Example 4.8. Let G , denote the Grassmannian of lines in 3-space, which is the varietyin P C , defined by P P − P P + P P = 0 , which we consider as a subvariety of ( C ∗ ) . The three terms in this quadratic equationinvolve distinct variables and hence correspond to linearly independent exponent vectors.Note that G , equals γ − ( V ) where γ : ( C ∗ ) → ( C ∗ ) , ( p , p , p , p , p , p ) ( p p , p p , p p )and V denotes the plane in 3-space defined by the linear equation X − Y + Z = 0. Sinceby Theorem 4.6 any point in the complement has a certificate of degree 2, Theorem 4.7 PPROXIMATING AMOEBAS AND COAMOEBAS 15 −3 −2 −1 0 1 2 3 4−3−2−101234 Figure 4. SOS certificates of linear amoebas restricted to degree two. Thered (dark) points represent infeasible SDPs, and in the green (light) pointsnumerical instabilities were reported in the computations.implies that every point in the complement of the Grassmannian amoeba has a certificateof degree 4.5. Computing the relaxations via semidefinite programming We close the paper by providing some computational results in order to confirm thevalidity of our approach. The subsequent computations have been performed on topof SOSTools [18] which is a Matlab package for computing sums of squares basedcomputations. The SDP package underlying SOSTools is SeDuMi [23]. Example 5.1. For the test case of a linear polynomial f := Z + 2 Z + 3, the boundarycontour of the amoeba A f can be explicitly described, and it is given by the curvesexp( z ) = 2 · exp( z ) + 3 , · exp( z ) = exp( z ) + 3 , z ) + 2 · exp( z ) , see [2]. We compute the amoeba of f with our SDP via SOSTools on a grid of size250 × 250 lattice points in the area [ − , . In the SDP we restrict to polynomials ofdegree two. By Theorem 4.4, the approximation is exact in that case (up to numericalissues). Figure 4 visualizes the SDP-based computation of the SOS certificates. In thefigure, at the white outer regions, certificates are found. At the dark red points the SDPis infeasible; at the green points, no feasibility is proven, but numerical instabilities arereported by the SDP solver. The aspect of numerical stability of our SOS-based amoebacomputations and of general SOS computations is an important issue in convex algebraicgeometry which deserves further study. See [7, 25] for existing work in this direction,based on choosing different bases. Example 5.2. As in Figure 1 we consider the class of polynomials f := Z Z + Z Z + c · Z Z + 1 with some constant c ∈ R . We use the monomial-based approach fromCorollary 3.7. In order to compute whether a given point ( λ , λ ) ∈ (0 , ∞ ) is containedin the unlog amoeba U f , we have to consider the polynomials h = ( X X − X Y Y − Y · X ) + ( X X − X Y − Y X Y )+ c · ( X X − Y Y ) + 1 ,h = ( X Y + 2 X Y X − Y Y ) + (2 X X Y + Y X − Y Y ) + c · ( X Y + X Y ) ,h = ( X X − X Y Y − Y X ) + ( X Y + 2 X Y X − Y Y ) − ( λ · λ ) ,h = ( X X − X Y − Y X Y ) + (2 X X Y + Y X − Y Y ) − ( λ · λ ) ,h = ( X X − Y Y ) + ( X Y + X Y ) − ( λ · λ ) . For the case c = 2 and c = − × 160 points in the area [ − , × [ − , A f is depicted in Figure5. At the white points the SDP is feasible and thus these points belong to the complementcomponent. At the orange points the SDP is recognized as feasible with numerical issues(within a pre-defined range). At the black points in the center the SDP was infeasiblewithout and at the turquoise points with numerical issues reported. At the red points inthe upper right corner the program stopped due to exceeding numerical problems. Theunion of the central black, the turquoise and part of the orange points provides the (degreebounded) approximation of the amoeba. Figure 5. The amoeba of f := Z Z + Z Z + c · Z Z + 1 approximatedwith SOStools for c = 2 and c = − 4. See the text of Example 5.2 for anexplanation of the colors and an adress of the numerical issues. The diameter of inner complement components. We briefly discuss that the SOS-based certificates can also be used for more sophisticated questions rather than the puremembership problem. For this, we consider a family of polynomials f ∈ C [ Z , . . . , Z n ]whose Newton polytope is a simplex and which have n + 2 monomials such that one of PPROXIMATING AMOEBAS AND COAMOEBAS 17 them is located in the interior of the simplex. Amoebas of polynomials of this class haveat most one inner complement component (see [27] for a comprehensive investigation ofthat class).Let f := P n +1 i =0 b i · Z α ( i ) , and let b denote the coefficient of the inner monomial. Bythe results in [27], for | b | → ∞ the inner complement component appears at the imageunder the Log–map of a minimal point δ of the function ˆ f := (cid:12)(cid:12)(cid:12) f arg( b ) · Z α (0) (cid:12)(cid:12)(cid:12) . δ is explicitlycomputable, and | δ | is unique. This allows to certify that a complement component ofthe unlog amoeba has a certain diameter d under the scaling | Z | 7→ | Z | of the (unlog)amoeba basis space by solving the SDP corresponding to X j =1 s j g j + H + 1 = 0with s = P ni =1 ( | δ i | − | Z i | ) − d / s = P n +1 i =0 (cid:0) b i · Z α ( i ) (cid:1) re and s = P n +1 i =0 (cid:0) b i · Z α ( i ) (cid:1) im ,where g j ∈ C [ Z ] (restricted to some total degree) and H is an SOS polynomial.Feasibility of the SDP certifies that there exists no point v ∈ V ( f ) ∩ ∂ B d/ ( δ ) (where B d/ ( δ ) denotes the ball with radius d/ δ ) in the rescaled amoeba basis space.Hence, the corresponding inner complement component of the unlog amoeba has at leasta diameter d in that space. We have to investigate the rescaled basis space of the unlogamoeba in order to transform the generic condition ( | δ i | − | Z i | ) − d / U f into a polynomial condition, which is given by s here.Note that this works not only for polynomials in the class under investigation, but forevery polynomial as long as one knows, where a complement component appears. Example 5.3. As before, let f := Z Z + Z Z + c · Z Z + 1 with a real parameter c . For this class, the inner complement component appears at the point (1 , 1) and thusunder the Log-map at the origin of Log( R ). The inner complement component existsfor c > c < − − , − 9] and [1 , 7] withsteplength 0 . 1. For any of these points we compute 14 SDPs in order to estimate theradius (based on binary search). In the rescaled amoeba basis space we obtain the boundsshown in Figure 6.Observe that these bounds are lower bounds since feasibility of the SDP certifies mem-bership in the complement of the amoeba but infeasibility only certifies that no certificatewith polynomials of degree at most k (i.e., 3 in our case) exists.This approach also yields lower bounds for the diameter of the inner complement com-ponent of the (log) amoeba. The image of the circle P ni =1 ( | δ i | −| Z i | ) − r under rescalingand the Log–map, i.e. P ni =1 (log | δ i | − log | Z i | ) − r , contains the set of points n ( | δ | · e r , . . . , | δ n | ) , . . . , ( | δ | , . . . , | δ n | · e r ) , (cid:16) | δ | · e r √ n , . . . , | δ n | · e r √ n (cid:17)o . By convexity of the complement components in A f , the simplex spanned by these pointsis contained in the inner complement component. Hence the double radius of the insphere −9 −8 −7 −6 −5 −4 −300.20.40.60.811.21.41.61.8 1 2 3 4 5 6 700.20.40.60.811.21.41.61.82 Figure 6. Lower bound for the diameter of the inner complement component.of that simplex is a lower bound for the diameter of the inner complement componentof A f . 6. Outlook We have developed foundations and techniques for approximating amoebas and coamoe-bas based on the real Nullstellensatz and sums-of-squares techniques. While our focus wason developing the core principles of the computational methodology, some experimentalresults were provided to show the validity of the approach. Beyond the specific resultswe have presented, our approach can be seen as a first systematic treatment of amoebasfrom a (computational) real algebraic point of view and we think that this viewpoint willhave more potential to offer.Major current challenges involve both computational issues (concerning the qualityand efficiency of computations) as well theoretical questions on on the real algebraicviewpoint on amoebas. To name a specific open question in the latter respect, recall thatin Theorem 4.4, for a hypersurface amoeba A f we could deduce the order of a complementcomponent from the special certificates we treated in that theorem. However, it is an openand important structural question how to deduce the order for a point w in the complementof A f given an arbitrary Nullstellensatz certificate for w . Acknowledgment. Thanks to Mihai Putinar for pointing out reference [10] and to ananonymous referee for careful reading and helpful suggestions. 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