Computing real radicals by moment optimization
aa r X i v : . [ m a t h . A C ] F e b Computing real radicals by moment optimization
Lorenzo Baldi & Bernard Mourrain
Inria Méditerranée, Université Côte d’Azur,Sophia Antipolis, France
ABSTRACT
We present a new algorithm for computing the real radical of anideal 𝐼 and, more generally, the 𝑆 -radical of 𝐼 , which is based onconvex moment optimization. A truncated positive generic linearfunctional 𝜎 vanishing on the generators of 𝐼 is computed solvinga Moment Optimization Problem (MOP). We show that, for a largeenough degree of truncation, the annihilator of 𝜎 generates the realradical of 𝐼 . We give an effective, general stopping criterion on thedegree to detect when the prime ideals lying over the annihilatorare real and compute the real radical as the intersection of realprime ideals lying over 𝐼 .The method involves several ingredients, that exploit the prop-erties of generic positive moment sequences. A new efficient algo-rithm is proposed to compute a graded basis of the annihilator of atruncated positive linear functional. We propose a new algorithmto check that an irreducible decomposition of an algebraic varietyis real, using a generic real projection to reduce to the hypersur-face case. There we apply the Sign Changing Criterion, effectivelyperformed with an exact MOP. Finally we illustrate our approachin some examples. KEYWORDS real radical, moments, positive polynomial, convex optimization,orthogonal polynomials, numerical algorithm
ACKNOWLEDGMENTS
This work is supported by the European Union’s Horizon 2020 re-search and innovation programme under the Marie Skłodowska-Curie Actions, grant agreement 813211 (POEMA).
In many “real world” problems which can be modeled by poly-nomial constraints, the solutions with real coordinates are gener-ally analyzed with particular attention. Efficient algebraic methodshave been developed over the years to solve such systems of poly-nomial constraints, including Grobner basis, border basis, resul-tants, triangular sets, homotopy continuation. But all these meth-ods involve implicitly the complex roots of the polynomial systemsand their complexity depends on the degree (and multiplicity) ofthe underlying complex algebraic varieties.Finding equations vanishing on the real solutions without com-puting all the complex roots is a challenging question. This meanscomputing the vanishing ideal of the real solutions of an ideal 𝐼 ,that is, its real radical R √ 𝐼 .Several approaches have been proposed to compute the real rad-ical. Some of these methods are reducing to univariate problems[5, 6, 29, 36], or exploiting quantifier elimination techniques [15],or using infinitesimals [31] or triangular sets and regular chains[12, 39]. Sums-of-Squares convex optimisation and moment matrices areused in [20, 21] to compute real radicals, when the set of real so-lutions is finite. Some properties of ideals associated to semidefi-nite programming relaxations are analysed in [33], involving thesimple point criterion. In [25] a stopping criterion is presented toverify that a Pommaret basis has been computed from the kernelsof moment matrices involved in Sum of Squares relaxation. In [10],a test based on sum-of-square decomposition is proposed to verifythat polynomials vanishing on a subset of the semi-algebraic setare in the real radical.In [32], an algorithm based on rational representations of equidi-mensional components of algebraic varieties and singular locus re-cursion is presented and its complexity is analysed.We present a new algorithm for computing the real radical ofan ideal 𝐼 and, more generally, the 𝑆 -radical of 𝐼 , which is basedon convex moment optimization. An interesting feature of the ap-proach is that it does not involve the complex solutions, which arenot on a real component of the algebraic variety V ( 𝐼 ) . Section 2recalls the relationship between vanishing ideals and radicals forreal and complex algebraic varieties.Generators of the real radical of 𝐼 are computed from a truncatedgeneric positive linear functional 𝜎 vanishing on the generators of 𝐼 . This truncated linear functional is computed by solving a Mo-ment Optimization Problem (MOP), as summarized in Section 3.We show that, for a large enough degree of truncation, the an-nihilator of 𝜎 generates the real radical of 𝐼 , suggesting an algo-rithm which will compute the annihilator of a generic positive lin-ear functional for increasing degrees. The flat extension property(see e.g. [14, 22]) gives a stopping criterion in the zero-dimensionalcase (see e.g. [20, 21]). But the question remained open for positive-dimensional real varieties (see e.g. [23, § 4.3], [25]).In this work, we give a new effective stopping criterion to detectwhen the prime ideals associated to the annihilator are real andcompute equations for the minimal real prime ideals lying over 𝐼 .The method involves several ingredients, that exploit the prop-erties of generic non-negative moment sequences.A new efficient algorithm is proposed in Section 4 to compute agraded basis of the annihilator of a truncated non-negative linearfunctions. A new algorithm is presented in Section 5 to check thatan irreducible decomposition of an algebraic variety is real, usinga generic real projection to reduce to the hypersurface case. Weapply the Sign Changing Criterion, effectively performed with anexact MOP.The complete algorithm for computing the real radical of anideal 𝐼 as the intersection of real prime ideals is presented in Sec-tion 6.In Section 7, we illustrate the algorithm by some effective nu-merical computation on examples, where the real radical differssignificantly from the ideal 𝐼 . orenzo Baldi & Bernard Mourrain Let 𝑓 , . . . , 𝑓 𝑠 ∈ C [ 𝑥 , . . . , 𝑥 𝑛 ] = C [ x ] and let 𝐼 = ( f ) ⊂ C [ x ] bethe ideal generated f = { 𝑓 , . . . , 𝑓 𝑠 } . The algebraic variety definedby f is denoted 𝑉 = V C ( 𝐼 ) = { 𝜉 ∈ C 𝑛 | 𝑓 𝑖 ( 𝜉 ) = , 𝑖 = , . . . , 𝑠 } . Itdecomposes into an union of irreducible components 𝑉 = ∪ 𝑙𝑖 = 𝑉 𝑖 where 𝑉 𝑖 = V C ( 𝔭 𝑖 ) with 𝔭 𝑖 a prime ideal of C [ x ] . An irreduciblevariety 𝑉 is an algebraic variety which cannot be decomposed intoan union of algebraic varieties distinct from 𝑉 .The Hilbert Nullstellensatz states that the vanishing ideal I ( 𝑉 ) = { 𝑝 ∈ C [ x ] | ∀ 𝜉 ∈ 𝑉 , 𝑝 ( 𝜉 ) = } of an algebraic variety 𝑉 ⊂ C 𝑛 isthe radical √ 𝐼 = { 𝑝 ∈ C [ x ] | ∃ 𝑚 ∈ N , 𝑝 𝑚 ∈ 𝐼 } (see e.g. [13]). This implies that √ 𝐼 = ∩ 𝑙𝑖 = 𝔭 𝑖 . We say that 𝐼 is radicalif 𝐼 = √ 𝐼 .Considering now equations f = { 𝑓 , . . . , 𝑓 𝑠 } ⊂ R [ x ] with realcoefficients, the real variety defined by f is 𝑉 R = V C ( 𝐼 ) ∩ R 𝑛 = V R ( 𝐼 ) = { 𝜉 ∈ R 𝑛 | 𝑓 𝑖 ( 𝜉 ) = , 𝑖 = , . . . , 𝑠 } . The vanishing idealof 𝑉 R is I ( 𝑉 R ) = { 𝑝 ∈ R [ x ] | ∀ 𝜉 ∈ 𝑉 R , 𝑝 ( 𝜉 ) = } . Let Σ = { Í 𝑗 𝑝 𝑗 , 𝑝 𝑗 ∈ R [ x ]} be the sums of squares of polynomials of R [ x ] .The real Nullstellensatz states that I (V R ( 𝐼 )) is the real radical of 𝐼 , defined as: R √ 𝐼 = { 𝑝 ∈ R [ x ] | ∃ 𝑚 ∈ N , 𝑠 ∈ Σ s.t. 𝑝 𝑚 + 𝑠 ∈ 𝐼 } (see e.g. [26, p. 26], [8, p. 85]). If 𝐼 = R √ 𝐼 then we say that 𝐼 is a real or real radical ideal. The real radical of 𝐼 contains √ 𝐼 and isthe intersection of real prime ideals 𝔭 𝑖 in R [ x ] containing 𝐼 , corre-sponding to the real irreducible components of V R ( 𝐼 ) . The exam-ple 𝑓 = 𝑥 + 𝑥 such that 𝐼 = ( 𝑓 ) = √ 𝐼 and R √ 𝐼 = ( 𝑥 , 𝑥 ) showsthat the radical and real radical ideals can define algebraic varietiesof different dimensions.Sets 𝑆 = { 𝜉 ∈ R 𝑛 | 𝑓 ( 𝜉 ) = , . . . , 𝑓 𝑠 ( 𝜉 ) = , 𝑔 ( 𝜉 ) ≥ , . . . ,𝑔 𝑟 ( 𝜉 ) ≥ } with 𝑓 𝑖 , 𝑔 𝑗 ∈ R [ x ] are called basic semi-algebraic sets.The real Nullstellensatz for 𝑆 states that the vanishing ideal I( 𝑆 ) is the 𝑆 -radical of 𝐼 = ( f ) : 𝑆 √ 𝐼 = { 𝑝 ∈ R [ x ] | ∃ 𝑚 ∈ N , ( 𝑠 𝛼 ) ∈ ( Σ ) { , } 𝑟 s.t. 𝑝 𝑚 + Õ 𝛼 𝑠 𝛼 g 𝛼 ∈ 𝐼 } (see e.g. [26, th. 2.2.1], [8, cor. 4.4.3], [17], [37]). The 𝑆 -radical 𝑆 √ 𝐼 isrelated to the real radical of an extended ideal 𝐼 𝑆 defined by intro-ducing slack variables 𝑠 , . . . , 𝑠 𝑟 for each non-negativity constraintdefining 𝑆 : 𝐼 𝑆 = ( 𝑓 , . . . , 𝑓 𝑠 , 𝑔 − 𝑠 , . . . , 𝑔 𝑟 − 𝑠 𝑟 ) ⊂ R [ 𝑥 , . . . , 𝑥 𝑛 , 𝑠 ,. . . , 𝑠 𝑟 ] . Namely, we have 𝑆 √ 𝐼 = R √ 𝐼 𝑆 ∩ R [ x ] (by the Real Nullstellen-satz, see e.g. [8, p. 91]). Therefore, in the following we will focuson the computation of the real radical of 𝐼 = ( f ) and apply thistransformation for the computation of 𝑆 -radicals.To describe the irreducible components of a variety V C ( 𝑓 , . . . , 𝑓 𝑠 ) defined by equations 𝑓 , . . . , 𝑓 𝑠 ∈ R [ x ] , we use tools from Numer-ical Algebraic Geometry, namely a description of irreducible com-ponents by witness sets. A witness set of an irreducible algebraicvariety 𝑉 ⊂ C 𝑛 is a triple 𝑊 = ( f , 𝐿, 𝑆 ) where f ⊂ I( 𝑉 ) , 𝐿 is ageneric linear space of dimension 𝑛 − dim ( 𝑉 ) given by dim ( 𝑉 ) linear equations and 𝑆 = 𝐿 ∩ 𝑉 ⊂ C 𝑛 is a finite set of deg ( 𝑉 ) points. Given equations f = { 𝑓 , . . . , 𝑓 𝑠 } ⊂ R [ x ] , a numerical irre-ducible decomposition of V C ( f ) can be computed as a collectionof witness sets 𝑊 𝑖 = ( h 𝑖 , 𝐿 𝑖 , 𝑆 𝑖 ) such that each irreducible compo-nent 𝑉 𝑖 of 𝑉 is described by one and only one witness set 𝑊 𝑖 and all sample sets 𝑆 𝑖 are pairewise disjoint. Several methods, based on ho-motopy techniques, have been developed over the past to computesuch decomposition. See e.g. [4, 16, 35].The witness set 𝑊 of an (irreducible) algebraic variety 𝑉 , can beused to compute defining equations h = { ℎ , . . . , ℎ 𝑛 } ⊂ C [ x ] suchthat V C ( h ) = 𝑉 . Homotopy techniques are employed to generateenough sample points on 𝑉 . The equations ℎ 𝑖 are then computedby projection of the sample points onto ≤ 𝑛 + generic linearspaces of dimension ( dim ( 𝑉 ) + ) and by interpolation. See e.g.[35], for more details.The numerical irreducible decomposition of V C ( f ) as a collec-tion of witness sets provides a description of all the irreduciblecomponents 𝑉 𝑖 associated to the isolated primary components 𝑄 𝑖 of 𝐼 = ( f ) [2]. To check that these primary components are re-duced and thus prime (i.e. √ 𝑄 𝑖 = 𝑄 𝑖 ), it is enough to check thatthe Jacobian of f is of rank 𝑛 − dim 𝑉 𝑖 (Jacobian criterion) at one ofthe sample points of the witness set 𝑊 𝑖 , describing the irreduciblecomponent 𝑉 𝑖 = V ( 𝑃 𝑖 ) .Checking that 𝐼 = ( f ) has no embedded component can also bedone by numerical irreducible decomposition of deflated ideals, asdescribed in [18]. We are not going to use this deflation techniqueto check non-embedded components. We describe the dual of polynomial rings (see for instance [28] formore details). For 𝜎 ∈ ( R [ x ]) ∗ = hom R ( R [ x ] , R ) = { 𝜎 : R [ x ] → R | 𝜎 is R -linear } , we denote h 𝜎, 𝑓 i = 𝜎 ( 𝑓 ) the application of 𝜎 to 𝑓 ∈ R [ x ] . Recall that ( R [ x ]) ∗ (cid:27) R [[ y ]] ≔ R [[ 𝑦 , . . . , 𝑦 𝑛 ]] , withthe isomorphism given by: 𝜎 ↦→ Í 𝛼 ∈ N 𝑛 h 𝜎, x 𝛼 i y 𝛼 𝛼 ! , where { y 𝛼 𝛼 ! } is the dual basis to { x 𝛼 } , i.e. h y 𝛼 , x 𝛽 i = 𝛼 ! 𝛿 𝛼,𝛽 . With this basiswe can also identify 𝜎 ∈ ( R [ x ]) ∗ with its sequence of coefficients ( 𝜎 𝛼 ) 𝛼 , where 𝜎 𝛼 ≔ h 𝜎, x 𝛼 i .If 𝜎 ∈ ( R [ x ]) ∗ and 𝑔 ∈ R [ x ] , we define the convolution of 𝑔 and 𝜎 as 𝑔 ★ 𝜎 ≔ 𝜎 ◦ 𝑚 𝑔 ∈ ( R [ x ]) ∗ where 𝑚 𝑔 is the operator ofmultiplication by 𝑔 on the polynomials (i.e. h 𝑔 ★ 𝜎, 𝑓 i = h 𝜎, 𝑔𝑓 i ∀ 𝑓 ).The operation ★ defines an R [ x ] -module structure on R [[ y ]] . Wedefine the Hankel operator 𝐻 𝜎 : R [ x ] → ( R [ x ]) ∗ , 𝑔 ↦→ 𝑔 ★ 𝜎 andthe annihilator Ann ( 𝜎 ) = ker 𝐻 𝜎 : 𝑔 ∈ Ann ( 𝜎 ) ⇐⇒ 𝐻 𝜎 ( 𝑔 ) = ⇐⇒ 𝑔 ★ 𝜎 = .We describe these operations in coordinates. If 𝜎 = ( 𝜎 𝛼 ) 𝛼 and 𝑔 = Í 𝛼 𝑔 𝛼 x 𝛼 then 𝑔 ★ 𝜎 = ( Í 𝛽 𝑔 𝛽 𝜎 𝛼 + 𝛽 ) 𝛼 ; the matrix 𝐻 𝜎 in thebasis { x 𝛼 } and { y 𝛼 𝛼 ! } is 𝐻 𝜎 = ( 𝜎 𝛼 + 𝛽 ) 𝛼,𝛽 . We introduce the same operations in a finite dimensional setting,considering only polynomials of bounded degree. If 𝐴 ⊂ R [ x ] , 𝐴 𝑑 ≔ { 𝑓 ∈ 𝐴 | deg 𝑓 ≤ 𝑑 } . In particular R [ x ] 𝑑 is the vec-tor space of polynomials of degree ≤ 𝑑 . If 𝜎 ∈ ( R [ x ]) ∗ (resp. 𝜎 ∈ ( R [ x ] 𝑟 ) ∗ , 𝑟 ≥ 𝑡 ) then 𝜎 [ 𝑡 ] ∈ ( R [ x ] 𝑡 ) ∗ denotes its restrictionto R [ x ] 𝑡 ; moreover if 𝐵 ⊂ ( R [ x ]) ∗ (resp. 𝐵 ⊂ ( R [ x ] 𝑟 ) ∗ , 𝑟 ≥ 𝑡 )then 𝐵 [ 𝑡 ] ≔ { 𝜎 [ 𝑡 ] ∈ ( R [ x ] 𝑡 ) ∗ | 𝜎 ∈ 𝐵 } .If 𝜎 ∈ ( R [ x ] 𝑡 ) ∗ and 𝑔 ∈ R [ x ] 𝑡 , then 𝑔 ★ 𝜎 ≔ 𝜎 ◦ 𝑚 𝑔 ∈( R [ x ] 𝑡 − deg 𝑔 ) ∗ . If 𝜎 ∈ ( R [ x ]) ∗ (or 𝜎 ∈ ( R [ x ] 𝑟 ) ∗ , 𝑟 ≥ 𝑡 ), thenwe define 𝐻 𝑡𝜎 : R [ x ] 𝑡 → ( R [ x ] 𝑡 ) ∗ , 𝑔 ↦→ ( 𝑔 ★ 𝜎 ) [ 𝑡 ] . We have omputing real radicals by moment optimization ( 𝑔 ★ 𝜎 ) [ 𝑡 ] = ⇐⇒ 𝐻 𝑡𝑔 ★ 𝜎 = : in analogy to the infinite di-mensional setting we define Ann 𝑑 ( 𝜎 ) ≔ ker 𝐻 𝑑𝜎 .For h = ℎ , . . . , ℎ 𝑟 ⊂ R [ x ] we define h h i 𝑡 ≔ (cid:8) Í 𝑟𝑖 = 𝑓 𝑖 ℎ 𝑖 ∈ R [ X ] 𝑡 | 𝑓 𝑖 ∈ R [ X ] 𝑡 − deg ℎ 𝑖 (cid:9) , the elements of ( h ) 𝑡 generated indegree ≤ 𝑡 . Let 𝐴 ⊂ R [ x ] (resp. 𝐴 ⊂ R [ x ] 𝑡 ). We define 𝐴 ⊥ ≔ (cid:8) 𝜎 ∈ ( R [ x ]) ∗ |h 𝜎, 𝑓 i = ∀ 𝑓 ∈ 𝐴 (cid:9) (resp. 𝐴 ⊥ ≔ (cid:8) 𝜎 ∈ ( R [ x ] 𝑡 ) ∗ | h 𝜎, 𝑓 i = ∀ 𝑓 ∈ 𝐴 (cid:9) ). Notice that 𝜎 ∈ h h i ⊥ 𝑡 (resp. ( h ) ⊥ ) if and only if ( ℎ ★ 𝜎 ) [ 𝑡 − deg ℎ ] = ∀ ℎ ∈ h (resp. ℎ ★ 𝜎 = ∀ ℎ ∈ h ).We say that 𝜎 ∈ ( R [ x ] 𝑡 ) ∗ is positive semidefinite (psd) ⇐⇒ 𝐻 𝑡𝜎 is psd, i.e. h 𝐻 𝑡𝜎 ( 𝑓 ) , 𝑓 i = h 𝜎, 𝑓 i ≥ ∀ 𝑓 ∈ R [ x ] 𝑡 . If 𝜎 is pdsthen h 𝜎, 𝑓 i = ⇒ 𝑓 ∈ Ann 𝑡 ( 𝜎 ) [20, 3.12]For 𝐺 ⊂ R [ x ] 𝑡 we finally define the closed convex cone: L 𝑡 ( 𝐺 ) ≔ { 𝜎 ∈ ( R [ x ] 𝑡 ) ∗ | 𝜎 is psd and ∀ 𝑔 ∈ ( 𝐺 · Σ ) 𝑡 h 𝜎, 𝑔 i ≥ } , see [3] for more details. In particular L 𝑡 (± h ) = { 𝜎 ∈ h h i ⊥ 𝑡 | 𝜎 is psd } . We use L( 𝐺 ) for the infinite dimensional case. Noticethat L( 𝐺 ) [ 𝑡 ] ⊂ L 𝑡 ( 𝐺 ) ∀ 𝑡 .Linear functionals of special importance are evaluations e 𝜉 de-fined as h e 𝜉 , 𝑓 i = 𝑓 ( 𝜉 ) . For 𝜉 ∈ V R ( h ) we have e 𝜉 ∈ L(± h ) . Definition 3.1.
We say that 𝜎 ∗ ∈ L 𝑡 (± h ) is generic if rank 𝐻 𝑡𝜎 ∗ = max { rank 𝐻 𝑡𝜂 | 𝜂 ∈ L 𝑡 (± h )} .Genericity is characterized as follows, see e.g. [20, prop. 4.7]: Proposition 3.2.
Let 𝜎 ∈ L 𝑡 (± h ) . The following are equiva-lent:(i) 𝜎 is generic;(ii) Ann 𝑡 ( 𝜎 ) ⊂ Ann 𝑡 ( 𝜂 ) ∀ 𝜂 ∈ L 𝑡 ( g ) ;(iii) ∀ 𝑑 ≤ 𝑡 , we have: rank 𝐻 𝑑𝜎 = max { rank 𝐻 𝑑𝜂 | 𝜂 ∈ L 𝑡 (± h )} . Generic elements can be used to compute the real radical ofideals, see [30, th. 7.39]. We give in Theorem 3.3 a proof of thisresult. See also [3, th. 3.16] for a generalisation to quadratic mod-ules.
Theorem 3.3.
Let 𝜎 ∗ ∈ L 𝑑 (± h ) be generic and 𝐼 = ( h ) . Thenfor every 𝑑 ≥ deg h we have 𝐼 ⊂ ( Ann 𝑑 ( 𝜎 ∗ )) ⊂ R √ 𝐼 . Moreover for 𝑑 big enough ( Ann 𝑑 ( 𝜎 ∗ )) = R √ 𝐼 . Proof.
The inclusion 𝐼 ⊂ ( Ann 𝑑 ( 𝜎 ∗ )) is clear since h ⊂ Ann 𝑑 ( 𝜎 ∗ ) by definition. Now let 𝐽 = R √ 𝐼 . Notice that, for 𝜉 ∈ R 𝑛 , Ann 𝑑 ( e 𝜉 ) = I ( 𝜉 ) 𝑑 = ( 𝑥 − 𝜉 , . . . , 𝑥 𝑛 − 𝜉 𝑛 ) 𝑑 . Moreover, if 𝜉 ∈ V R ( 𝐼 ) , then e [ 𝑑 ] 𝜉 ∈ L 𝑑 (± h ) . Then, since 𝜎 ∗ is generic: Ann 𝑑 ( 𝜎 ∗ ) ⊂ Ù 𝜉 ∈V R ( 𝐼 ) Ann 𝑑 ( e 𝜉 ) = Ù 𝜉 ∈V R ( 𝐼 ) I( 𝜉 ) 𝑑 = 𝐽 𝑑 , and thus ( Ann 𝑑 ( 𝜎 ∗ )) ⊂ 𝐽 .For the second part, let 𝑔 , . . . , 𝑔 𝑘 be generators of 𝐽 . By theReal Nullstellensatz, ∀ 𝑖 there exists 𝑚 𝑖 ∈ N , 𝑠 𝑖 ∈ Σ such that 𝑔 𝑚𝑖 𝑖 + 𝑠 𝑖 ∈ 𝐼 . Then for 𝑑 big enough and 𝜎 ∈ L 𝑑 (± h ) we have h 𝜎 [ 𝑑 ] , 𝑔 𝑚𝑖 𝑖 + 𝑠 𝑖 i = , thus h 𝜎 [ 𝑑 ] , 𝑔 𝑚𝑖 𝑖 i = and 𝑔 𝑖 ∈ Ann 𝑑 ( 𝜎 ) .This implies 𝐽 ⊂ ( Ann 𝑑 ( 𝜎 )) for all 𝜎 ∈ L 𝑑 (± h ) , and in particularfor 𝜎 = 𝜎 ∗ generic. (cid:3) The goal of the paper is to find an effective algorithm, based onTheorem 3.3, to compute R √ 𝐼 . In the case of a finite real variety, theflat extension criterion [20, 21] certifies that ( Ann 𝑑 ( 𝜎 ∗ )) = R √ 𝐼 forsome 𝑑 ∈ N . We will focus in the positive dimensional case, whensuch a criterion cannot apply. Let 𝑓 , g ∈ R [ x ] . The goal of Polynomial Optimization is to find: 𝑓 ∗ ≔ inf (cid:8) 𝑓 ( 𝑥 ) ∈ R | 𝑥 ∈ R 𝑛 , 𝑔 𝑖 ( 𝑥 ) ≥ for 𝑖 = , . . . , 𝑠 (cid:9) . (1)that is the infimum 𝑓 ∗ of the objective function 𝑓 on the basic semi-algebraic set 𝑆 ≔ { 𝑥 ∈ R 𝑛 | 𝑔 𝑖 ( 𝑥 ) ≥ for 𝑖 = , . . . , 𝑠 } . Inparticular we will consider the case of equalities ℎ 𝑖 = , obtainedas ± ℎ 𝑖 ≥ . To solve problem (1) Lasserre [19] proposed to usetwo hierarchies of finite dimensional convex cones depending onan order 𝑑 ∈ N . We describe the Moment Matrix hierarchy and theproperty of exactness , see [3] for more details. Definition 3.4.
We define the
MoM relaxation of order 𝑑 of prob-lem (1) as L 𝑑 ( g ) and the infimum: 𝑓 ∗ MoM ,𝑑 ≔ inf (cid:8) h 𝜎, 𝑓 i ∈ R | 𝜎 ∈ L 𝑑 ( g ) , h 𝜎, i = (cid:9) . (2)We will call Problem (2) a Moment Optimization Problem (MOP).It can be efficiently solved by semidefinite programming, using in-terior point methods. Taking 𝑓 = , these methods yield an interiorpoint of L 𝑑 ( g ) , that is a generic element 𝜎 ∗ in L 𝑑 ( g ) .Usually we are interested in minimizers of 𝑓 with bounded norm,i.e. minimizers in some closed ball defined by 𝑟 −k x k ≥ ( Archimedeancondition ). If the Archimedean condition and some regularity con-ditions at the minimizers of 𝑓 hold (known as Boundary HessianConditions or BHC), the MoM relaxation is exact : for some 𝑑 ∈ N the minimum is reached, i.e. 𝑓 ∗ = 𝑓 ∗ MoM ,𝑑 , and we can effectivelyrecover the minimizers (see [3, th. 4.8]). Using the flat extentioncriterion for the Hankel matrix 𝐻 𝑑𝜎 (associated to a minimizing mo-ment sequence 𝜎 ) we can effectively test exactness. As BHC holdgenerically, exactness is also generic (see [3, cor. 4.9]). To compute the real radical, we need to compute a basis of theannihilator of a truncated positive linear functional 𝜎 ∈ R [ x ] ∗ 𝑑 such that h 𝜎, 𝑝 i ≥ for 𝑝 ∈ R [ x ] 𝑑 . In this section, we describe anefficient algorithm to compute a basis of Ann 𝑑 ( 𝜎 ) = { 𝑝 ∈ R [ x ] 𝑑 | 𝑝 ★ 𝜎 = } = { 𝑝 ∈ R [ x ] 𝑑 | h 𝜎, 𝑝 i = } . It is a Gram-Schmidtorthogonalization process, using the inner product h· , ·i 𝜎 defined,for 𝑝, 𝑞 ∈ R [ x ] 𝑑 , by h 𝑝, 𝑞 i 𝜎 : = h 𝜎, 𝑝 𝑞 i . By ordering the monomials basis of R [ x ] 𝑑 and projecting suc-cessively a monomial x 𝛼 onto the space spanned by the previousmonomials, we construct monomial basis b = { x 𝛽 } of R [ x ] 𝑑 / Ann 𝑑 ( 𝜎 ) ,a corresponding basis of orthogonal polynomials p = ( 𝑝 𝛽 ) and abasis k = ( 𝑘 𝛾 ) of Ann 𝑑 ( 𝜎 ) . The orthogonal polynomials are suchthat h 𝑝 𝛽 , 𝑝 𝛽 ′ i 𝜎 = (cid:26) > if 𝛽 = 𝛽 ′ otherwise , and for all 𝛽,𝛾 , we have h 𝑝 𝛽 , 𝑘 𝛾 i 𝜎 = h 𝑘 𝛾 , 𝑘 𝛾 i 𝜎 = . orenzo Baldi & Bernard Mourrain To compute these polynomials, we use a projection defined onthe orthogonal of the space spanned by orthogonal polynomials p = [ 𝑝 , . . . , 𝑝 𝑙 ] such that h 𝑝 𝑖 , 𝑝 𝑖 i 𝜎 > and h 𝑝 𝑖 , 𝑝 𝑗 i 𝜎 = if 𝑖 ≠ 𝑗 ,as follows: for 𝑓 ∈ R [ x ] 𝑑 ,proj ( 𝑓 , p ) = 𝑓 − 𝑙 Õ 𝑖 = h 𝑓 , 𝑝 𝑖 i 𝜎 h 𝑝 𝑖 , 𝑝 𝑖 i 𝜎 𝑝 𝑖 . By construction, we have h proj ( 𝑓 , p ) , 𝑝 𝑖 i 𝜎 = for 𝑖 = , . . . , 𝑙 .In practice, the implementation of this projection is done by theso-called Modified Gram-Schmidt projection algorithm, which isknown to have a better numerical behavior than the direct Gram-Schmidt orthogonalization process [38][Lecture 8].To compute a basis of Ann 𝑑 ( 𝜎 ) , we choose a monomial order-ing ≺ compatible with the degree (e.g. the graded reverse lexico-graphic ordering) and build the list of monomials s of degree ≤ 𝑑 in increasing order for this ordering ≺ . Algorithm 4.1 choses in-crementally a new monomial in the list s and projects it on thespace spanned by the previous orthogonal polynomials. The newmonomials computed by the function next ( s , b , l ) are the lowestmonomials of s not in b and not divisible by a monomial of l . Algorithm 4.1:
Orthogonal polynomials and annihilatorof 𝜎 Input: a positive linear functional 𝜎 ∈ R [ x ] ∗ 𝑑 + . • Let b : = [] ; p : = [] ; k : = [] ; l = [] ; n : = [ ] ; s : = [ x 𝛼 , | 𝛼 | ≤ 𝑑 ] ; • while n ≠ ∅ do – for each x 𝛼 ∈ n ,(i) 𝑝 𝛼 : = proj ( x 𝛼 , p ) ;(ii) compute 𝑣 𝛼 = h 𝑝 𝛼 , 𝑝 𝛼 i 𝜎 ;(iii) if 𝑣 𝛼 ≠ thenadd x 𝛼 to b ; add 𝑝 𝛼 to p ;elseadd 𝑘 𝛼 : = 𝑝 𝛼 to k ; add x 𝛼 to l ;end; – n : = next ( s , b , d ) ; Output: • a basis k = [ 𝑘 𝛾 ] x 𝛾 ∈ l of the annihilator Ann 𝑑 ( 𝜎 ) and theirleading monomials l = [ x 𝛾 ] ; • a basis of orthogonal polynomials p = [ 𝑝 𝛽 𝑖 ] , m = [ 𝑚 𝛽 𝑖 ] ; • a monomial set b = [ x 𝛽 , . . . , x 𝛽 𝑟 ] .By construction, the vector space spanned by b and p are equalat each loop of the algorithm. As the function next ( s , b , l ) outputsthe least monomials in s , greater than b and not divisible by themonomials in l , the monomials in n are greater than the monomialsin b . Thus, the leading term of 𝑘 𝛾 ∈ k is x 𝛾 .Let k , l , p , b denote the output of Algorithm 4.1. For 𝛼 ∈ N 𝑛 ,let ( k ) (cid:22) 𝛼 be the vector space spanned by the elements of the form x 𝛿 𝑘 𝛾 with 𝛿 + 𝛾 (cid:22) 𝛼 . Similarly, p (cid:22) 𝛼 is the set of 𝑝 𝛽 ∈ p such that 𝛽 (cid:22) 𝛼 . We prove that k is a Grobner basis of Ann 𝑑 ( 𝜎 ) , that is anyelement of Ann 𝑑 ( 𝜎 ) reduces to by k : Proposition 4.1.
Let 𝜎 ∈ R [ x ] ∗ 𝑑 + , k , p be the output of Al-gorithm 4.1. For x 𝛼 ∈ ( l ) 𝑑 , i.e. divisible by a monomial in l and ofdegree | 𝛼 | ≤ 𝑑 , 𝑝 𝛼 = proj ( x 𝛼 , p (cid:22) 𝛼 ) is in ( k ) (cid:22) 𝛼 ⊂ Ann 𝑑 ( 𝜎 ) . Proof.
Let us prove it by induction on the ordering of 𝛼 . Thelowest element in ( l ) 𝑑 is a monomial x 𝛾 of l . As 𝑘 𝛾 = proj ( x 𝛾 , p 𝛽 (cid:22) 𝛾 ) is such that h 𝑘 𝛾 , 𝑘 𝛾 i 𝜎 = h 𝜎, 𝑘 𝛾 i = , 𝑘 𝛾 = proj ( x 𝛾 , p 𝛽 (cid:22) 𝛾 ) ∈ ( k ) (cid:22) 𝛾 ⊂ Ann 𝑑 ( 𝜎 ) . The induction hypothesis is true for the lowest mono-mial of ( l ) 𝑑 .Assume that it is true for x 𝛼 ′ ∈ ( l ) 𝑑 and let 𝛼 be the nextmonomial in ( l ) 𝑑 for the monomial ordering ≺ . Then, there ex-ists x 𝛼 ′′ ∈ ( l ) (cid:22) 𝛼 ′ and 𝑖 ∈ , . . . , 𝑛 such that 𝑥 𝑖 x 𝛼 ′′ = x 𝛼 . As 𝑝 𝛼 − 𝑥 𝑖 𝑝 𝛼 ′′ has a leading term smaller that x 𝛼 , it can be written asa linear combination of 𝑝 𝛼 ′ = proj ( x 𝛼 ′ , p ≺ 𝛼 ′ ) with 𝛼 ′ ≺ 𝛼 . Moreprecisely, we have 𝑝 𝛼 = 𝑥 𝑖 𝑝 𝛼 ′′ + Õ 𝛿 ≺ 𝛼, x 𝛿 ∈( l ) 𝑑 𝜆 𝛿 𝑝 𝛽 + Õ 𝛽 ≺ 𝛼, x 𝛽 ∈ b 𝜇 𝛽 𝑝 𝛽 , for some 𝜆 𝛿 , 𝜇 𝛽 ∈ R .By induction hypothesis, 𝑝 𝛼 ′′ , 𝑝 𝛿 ∈ ( k ) (cid:22) 𝛼 ′ ⊂ ( k ) (cid:22) 𝛼 ⊂ Ann 𝑑 ( 𝜎 ) .Moreover, as 𝑝 𝛼 ′′ ∈ Ann 𝑑 ( 𝜎 ) ⊂ Ann 𝑑 + ( 𝜎 ) , for any 𝑝 ∈ R [ x ] 𝑑 wehave h 𝑥 𝑖 𝑝 𝛼 ′′ , 𝑝 i 𝜎 = h 𝑝 𝛼 ′′ , 𝑥 𝑖 𝑝 i 𝜎 = . This shows that 𝑥 𝑖 𝑝 𝛼 ′′ ∈( k ) (cid:22) 𝛼 ∩ Ann 𝑑 ( 𝜎 ) .By definition of 𝑝 𝛼 = proj ( x 𝛼 , p ≺ 𝛼 ) , h 𝑝 𝛼 , 𝑝 𝛽 i 𝜎 = for x 𝛽 ∈ b ≺ 𝛼 so that 𝜇 𝛽 = h 𝑝 𝛼 ,𝑝 𝛽 i 𝜎 h 𝑝 𝛽 ,𝑝 𝛽 i 𝜎 = and 𝑝 𝛼 ∈ ( k ) (cid:22) 𝛼 ∩ Ann 𝑑 ( 𝜎 ) .As ( k ) (cid:22) 𝛼 = ( k ) (cid:22) 𝛼 ′ + h 𝑝 𝛼 i , we have ( k ) (cid:22) 𝛼 ⊂ Ann 𝑑 ( 𝜎 ) , whichproves the induction hypothesis for 𝛼 and concludes the proof ofthe proposition. (cid:3) This proposition explains why the function next ( s , b , l ) only out-puts the least monomial in s , greater than b and not divisible by themonomials in l . This algorithm is an optimization of Algorithm 4.1in [28] or Algorithm 3.2 in [27]. It strongly exploits the positivity ofthe linear functional 𝜎 and improves significantly the performance.We will illustrate its behavior in Section 7. Remark 4.2.
When the real variety V R ( f ) is finite, the flat exten-sion test on the rank of 𝐻 𝑘𝜎 can be replaced by testing that the set l ofinitial terms contains a power of each variable 𝑥 𝑖 . This is equivalentto the fact that R [ x ]/( k ) is finite dimensional or equivalently thatthe rank of 𝐻 𝑑𝜎 is constant for 𝑑 ≫ . We introduce an effective algorithm for testing real radicality inthe irreducible case.
Let C 𝑁 be the 𝑁 -dimensional affine space and C [ 𝑡 , . . . , 𝑡 𝑁 ] = C [ t ] be its coordinate (polynomial) ring. We say that a propertyholds generically in C 𝑁 if there exists finitely many nonzero poly-nomials 𝜙 , . . . , 𝜙 𝑙 ∈ C [ t ] such that, for 𝜉 ∈ C 𝑁 , when 𝜙 ( 𝜉 ) ≠ , . . . , 𝜙 𝑙 ( 𝜉 ) ≠ the property holds for 𝜉 .In particular we will consider linear maps 𝐴 ∈ hom C ( C 𝑛 , C 𝑘 + ) as elements in C 𝑛 ( 𝑘 + ) in the natural way, and thus talk about generic linear maps . We recall the definition of smooth zero . We refer to [34] for thecomplex case and to [26] for the real case. omputing real radicals by moment optimization We say that a variety 𝑉 ⊂ C 𝑛 is defined over R , if I ( 𝑉 ) is gener-ated by a family of polynomials with coefficient in R . For 𝐴 ⊂ C 𝑛 we denote by cl ( 𝐴 ) its Zariski closure.Hereafter K denotes a field of characteristic and K its algebraicclosure. Definition 5.1.
Let 𝐼 = ( 𝑓 , . . . , 𝑓 𝑚 ) ⊂ K [ x ] be a prime idealand 𝑉 = V K ( 𝐼 ) . We say that 𝜉 ∈ V K ( 𝐼 ) is a smooth zero of 𝐼 if rank Jac ( 𝑓 , . . . , 𝑓 𝑚 )( 𝜉 ) = 𝑛 − dim 𝑉 .For K = C the mapping 𝑉 ↦→ I C ( 𝑉 ) is a bijection betweenirreducible varieties in C 𝑛 and prime ideals. Moreover, for a primeideal 𝐼 , smooth zeros of 𝐼 and smooth points of V C ( 𝐼 ) coincide,and they are dense. On the other hand for K = R the mapping 𝑉 ↦→ I R ( 𝑉 ) is a bijection between irreducible varieties in R 𝑛 andprime ideals which are real radical. For prime ideals 𝐼 which arenot real radical, smooth zeros of 𝐼 are not dense in V R ( 𝐼 ) . Example 5.2.
Here are examples of reducible and irreducible al-gebraic varieties with dense complex smooth points but with noreal smooth point. • 𝐼 = ( 𝑥 + 𝑦 ) ⊂ R [ 𝑥, 𝑦 ] is a prime, non real radical ideal, as V R ( 𝐼 ) = {( , )} and R √ 𝐼 = ( 𝑥, 𝑦 ) . 𝐼 does not have smoothreal zeros. Notice that ( 𝑥 + 𝑦 ) ⊂ C [ 𝑥, 𝑦 ] is not prime, since 𝑥 + 𝑦 = ( 𝑥 + 𝑖𝑦 )( 𝑥 − 𝑖𝑦 ) . • 𝐼 = ( 𝑥 + 𝑦 + 𝑧 ) ⊂ R [ 𝑥, 𝑦, 𝑧 ] is a prime, non real radicalideal, as V R ( 𝐼 ) = {( , , )} and R √ 𝐼 = ( 𝑥, 𝑦, 𝑧 ) . 𝐼 does nothave smooth real zeros. In this case ( 𝑥 + 𝑦 + 𝑧 ) ⊂ C [ 𝑥, 𝑦, 𝑧 ] is prime, since 𝑥 + 𝑦 + 𝑧 is irreducible over C .We recall criterions for testing whether a prime ideal 𝐼 ⊂ R [ x ] is real radical or not. Theorem 5.3 (Simple Point Criterion [26, th. 12.6.1]).
Let 𝐼 be a prime ideal of R [ x ] . The following are equivalent: • 𝐼 is a real radical ideal; • 𝐼 = I (V R ( 𝐼 )) ; • cl (V R ( 𝐼 )) = V C ( 𝐼 ) ; • 𝐼 has a smooth real zero.Definition 5.4. If 𝑉 ⊂ C 𝑛 then 𝑉 R denotes the real points of 𝑉 ,i.e. 𝑉 R = 𝑉 ∩ 𝑉 = 𝑉 ∩ R 𝑛 .Let 𝑉 ⊂ C 𝑛 be an irreducible variety defined over R and 𝐼 ⊂ R [ x ] the ideal defined by its real generators. If follows from Theo-rem 5.3 that 𝑉 R = V R ( 𝐼 ) is Zariski dense in 𝑉 if and only if 𝐼 is areal radical ideal. In this case we say that 𝑉 is real .For hypersurfaces there exists another criterion based on thechange of sign of the defining polynomial. Theorem 5.5 (Sign Changing Criterion [26, th. 12.7.1]).
Let 𝑓 ∈ R [ x ] be an irreducible polynomial. The following are equivalent: • ( 𝑓 ) is a real radical ideal; • ( 𝑓 ) has a smooth real point (i.e. there exists 𝜉 ∈ V R ( 𝐼 ) suchthat ∇ 𝑓 ( 𝜉 ) ≠ ); • the polynomial 𝑓 changes sign in R 𝑛 (i.e. there exists 𝑥, 𝑦 ∈ R 𝑛 such that 𝑓 ( 𝑥 ) 𝑓 ( 𝑦 ) < ). We reduce the problem of testing real radicaly to the hypersurfacecase, and then use the Simple Point Criterion. For that prupose we project 𝑉 ⊂ C 𝑛 , irreducible variety of dimension 𝑘 , on a linearsubspace C 𝑘 + ⊂ C 𝑛 , in such a way 𝑉 and cl ( 𝜋 ( 𝑉 )) are birational .(see [34, p. 38] for the definition).It is classical that every irreducible (affine) variety is birationalto an hypersurface. We recall briefly this result to show that we canchoose a generic projection as birational morphism, as done for thegeometric resolution or rational representation, see for instance[24] or [9]. Lemma 5.6.
Let 𝑉 ⊂ C 𝑛 be an irreducible varierty of dimension 𝑘 and 𝜋 : C 𝑛 → C 𝑘 + be a generic projection. Then 𝑉 is birationalto 𝜋 ( 𝑉 ) , i.e. 𝑉 (cid:27) cl ( 𝜋 ( 𝑉 )) . Proof. (sketch) The birational morphism in [34, p. 39] can begiven as a generic projection. Indeed we can choose algebraicallyindependent elements 𝑙 , . . . , 𝑙 𝑘 generic linear forms in the inde-terminates x (see for instance [9, p. 488]). The choice of the prim-itive element 𝑙 𝑘 + is generic (see for instance [1, th. 15.8.1]: onecan choose 𝑙 𝑘 + as a generic linear form). Then 𝑙 , . . . , 𝑙 𝑘 + definethe projection 𝜋 : C 𝑛 → C 𝑘 + , 𝜉 ↦→ ( 𝑙 ( 𝜉 ) , . . . , 𝑙 𝑘 + ( 𝜉 )) and 𝑉 isbirational to cl ( 𝜋 ( 𝑉 )) . (cid:3) We choose a generic projection defined over R . In this case weshow that 𝑉 has a smooth real point if and only if cl ( 𝜋 ( 𝑉 )) has asmooth real point, using the following propositions. Proposition 5.7.
Let 𝑉 ⊂ C 𝑛 be an irreducible varierty definedover R of dimension 𝑘 , and let 𝜋 : C 𝑛 → C 𝑘 + be a generic projec-tion defined over R . Then cl ( 𝜋 ( 𝑉 )) is defined over R and if 𝑉 has asmooth real point then cl ( 𝜋 ( 𝑉 )) has a smooth real point. Proof.
Let 𝜋 : C 𝑛 → C 𝑘 + be a generic projection defined over R . As 𝑉 is defined over R , cl ( 𝜋 ( 𝑉 )) is also defined over R since I( 𝜋 ( 𝑉 )) is the elimination ideal (I ( 𝑉 ) + ( x − 𝜋 ( y )))∩ R [ y ] , where y = 𝑦 , . . . , 𝑦 𝑘 + are coordinates of C 𝑘 + (see [13]).If 𝑉 has a smooth real point then 𝑉 R is Zariski dense in 𝑉 byTheorem 5.3. Then 𝜋 ( 𝑉 R ) is Zariski dense in 𝜋 ( 𝑉 ) . Since 𝜋 is de-fined over R we have that 𝜋 ( 𝑉 R ) ⊂ ( 𝜋 ( 𝑉 )) R and ( 𝜋 ( 𝑉 )) R is Zariskidense in 𝜋 ( 𝑉 ) . Then cl (( 𝜋 ( 𝑉 )) R ) = cl ( 𝜋 ( 𝑉 )) and by Theorem 5.3 cl ( 𝜋 ( 𝑉 )) has a smooth real point. (cid:3) Proposition 5.8.
Let 𝑉 ⊂ C 𝑛 be an irreducible variety definedover R of dimension 𝑘 without smooth real points. Then, for a genericprojection 𝜋 : C 𝑛 → C 𝑘 + defined over R , cl ( 𝜋 ( 𝑉 )) is defined over R and has no smooth real points. Proof.
By Proposition 5.7, cl ( 𝜋 ( 𝑉 )) is defined over R .Assume now that cl ( 𝜋 ( 𝑉 )) has a smooth real point. Since 𝑉 is generically birational to 𝜋 ( 𝑉 ) (Lemma 5.6), the preimage of ageneric smooth point in 𝜋 ( 𝑉 ) is a single point in 𝑉 , which is smooth.If 𝜋 is defined over R then this smooth point 𝑝 ∈ 𝑉 is real since 𝜋 ( 𝑝 ) = 𝜋 ( 𝑝 ) = 𝜋 ( 𝑝 ) implies that 𝑝 = 𝑝 , showing that 𝑉 has asmooth real point. (cid:3) Proposition 5.9.
Let 𝑉 ⊂ C 𝑛 be an irreducible variety not de-fined over R of dimension 𝑘 . If 𝜋 : C 𝑛 → C 𝑘 + is a generic projectiondefined over R then cl ( 𝜋 ( 𝑉 )) is not defined over R . Proof. 𝑉 is not defined over R if and only if 𝑉 ≠ 𝑉 . Thus thereexists 𝑝 ∈ 𝑉 such that 𝑝 ∉ 𝑉 . Then for 𝜋 : C 𝑛 → C 𝑘 + a generic orenzo Baldi & Bernard Mourrain projection, we have 𝜋 ( 𝑝 ) ∉ cl ( 𝜋 ( 𝑉 )) (see e.g. [7, sec. 3]). As 𝜋 is defined over R , we have 𝜋 ( 𝑝 ) ∈ cl ( 𝜋 ( 𝑉 )) and 𝜋 ( 𝑝 ) = 𝜋 ( 𝑝 ) ∉ cl ( 𝜋 ( 𝑉 )) . Therefore, cl ( 𝜋 ( 𝑉 )) ≠ cl ( 𝜋 ( 𝑉 )) and cl ( 𝜋 ( 𝑉 )) is notdefined over R . (cid:3) Theorem 5.10.
Let 𝑉 ⊂ C 𝑛 be an irreducible variety of dimension 𝑘 . Then 𝑉 is defined over R and has a smooth real point if and onlyif, for 𝜋 : C 𝑛 → C 𝑘 + generic projection defined over R , cl ( 𝜋 ( 𝑉 )) isdefined over R and has a smooth real point. Proof. If 𝑉 has a smooth real point then we apply Proposi-tion 5.7 to conclude that cl ( 𝜋 ( 𝑉 )) has a smooth real point. If 𝑉 is defined over R but has no smooth real point, we apply Proposi-tion 5.8 and deduce that cl ( 𝜋 ( 𝑉 )) has no smooth real points. Fi-nally, if 𝑉 is not defined over R we apply Proposition 5.9 to showthat cl ( 𝜋 ( 𝑉 )) is not defined over R . (cid:3) Corollary 5.11.
Let 𝑉 ⊂ C 𝑛 be an irreducible variety of dimen-sion 𝑘 , and 𝜋 : C 𝑛 → C 𝑘 + a generic projection defined over R . Thenthe following are equivalent:(i) 𝑉 is defined over R and the real generators of I ( 𝑉 ) define areal radical ideal in R [ x ] ;(ii) I( 𝜋 ( 𝑉 )) is generated by a real polynomial, irreducible over C , which changes sign in R 𝑘 + . Proof.
By Theorem 5.3, real generators of
I ( 𝑉 ) define a realradical ideal if and only if 𝑉 has a smooth real point . Then ( 𝑖 ) ⇐⇒( 𝑖𝑖 ) follows from Theorem 5.10 and Theorem 5.5. (cid:3) We finally describe the algorithm for testing real radicality.
Algorithm 5.1:
Test real radicality
Input:
An irreducible variety 𝑉 ⊂ C 𝑛 of dim. 𝑘 and 𝜀, 𝑟 > .(i) Fix a generic projection 𝜋 : C 𝑛 → C 𝑘 + ;(ii) Compute the irreducible polynomial ℎ defining cl ( 𝜋 ( 𝑉 )) ;(iii) If ℎ is not real return false;(iv) Choose a generic point 𝜉 ∈ R 𝑘 + such that ℎ ( 𝜉 ) ≠ ;(v) 𝑠 : = sign ( ℎ ( 𝜉 )) ;(vi) Let 𝑓 = k x − 𝜉 k . Solve the MOP: 𝑓 ∗ MoM ,𝑑 = inf {h 𝜎, 𝑓 i | 𝜎 ∈ L 𝑑 (±( ℎ + 𝑠𝜀 ) , 𝑟 − 𝑓 ) , h 𝜎, i = } ; (vii) Extract a minimizer 𝜂 and check that ℎ ( 𝜉 ) ℎ ( 𝜂 ) < . Output:
False if the MOP is not feasible, true if the MOP isfeasible and ℎ ( 𝜉 ) ℎ ( 𝜂 ) < .In step (i) we fix a generic real projection such that 𝑉 is bira-tional to cl ( 𝜋 ( 𝑉 )) (Lemma 5.6).In steps (ii) and (iii) we compute a minimal degree polynomial ℎ of the hypersurface cl ( 𝜋 ( 𝑉 )) , scaled so that one of its coefficientsis and stop if it has non real coefficients.In steps (iv), (v) and (vi) we check if the real polynomial ℎ definesa real radical ideal, using Theorem 5.5. We find 𝜉 ∈ R 𝑘 + where ℎ isnot vanishing, and then search another point where ℎ has oppositesign, by Moment Optimization.If ℎ does not change sign then V R ( ℎ + 𝑠𝜀 ) = ∅ and the MOP willnot be feasible (see for instance [21]). On the other hand if ℎ changes sign there exist 𝜂 ∈ R 𝑘 + suchthat ℎ ( 𝜉 ) ℎ ( 𝜂 ) < . If k 𝜂 − 𝜉 k < 𝑟 and < 𝜀 ≤ 𝑓 ( 𝜂 ) then theMOP has a solution. For generic 𝜉 the minimizer will be a uniquesmooth point, the MOP will be exact (since we added the ball con-straint 𝑟 − 𝑓 ≤ , the Archimedean property holds and generecallythe MOM relaxation is exact), and we can certify that ℎ changessign. The constraint 𝑟 − k x − 𝜉 k ≥ is not necessary if V R ( ℎ ) iscompact, since in this case the Archimedean hypothesis is alreadysatisfied.The correctness of Algorithm 5.1 follows from Corollary 5.11. We test Algorithm 5.1 for two simple cases, using the Julia pack-ages
MomentTools.jl and
MultivariateSeries.jl . Example 5.12.
We check that the irreducible polynomial ℎ = 𝑥 + 𝑦 ∈ R [ 𝑥, 𝑦 ] defines an ideal 𝐼 = ( ℎ ) that is not real radical. Werandomly choose 𝜉 = (− . , − . ) ,where ℎ ( 𝜉 ) > . We check that ℎ does not change sign, detectingthe infeasibility of the optimization problem. X = @polyvar x yh = x^2 + y^2s = sign(f(X => xi))dist = sum((xi - vec(X)).^2)e = 0.01v, M = minimize(dist, [h+s*e], [9 - dist],X, 4, optimizer);
The termination status termination_status(M.model) of the op-timization:
INFEASIBLE::TerminationStatusCode = 2 shows the infeasibility of the moment optimization program andthat 𝐼 is not real radical.In the same way we detect the sign change. For ℎ = 𝑥 + 𝑦 − and 𝜉 as above, we find 𝜂 = (− . , − . ) and ℎ ( 𝜉 ) ℎ ( 𝜂 ) < .In the previous examples we could avoid the ball constraint 𝑟 − k x − 𝜉 k ≤ , since in these cases V R ( ℎ ) is compact and theArchimedean condition is already satisfied. With the main ingredients, we can now describe the algorithm forcomputing the real radical of an ideal 𝐼 = ( f ) , presented as the in-tersection of real prime ideals. The steps, summarised in Algorithm6.1, are detailed hereafter.In step (ii) we compute a generic element of L 𝑑 + (± f ) solvinga MOP with a constant objective function.In step (iii) we use Algorithm 4.1 to compute the graded basis k .In step (iv) we find the irreducible components of the variety V C ( k ) , described by witness sets (see e.g. [4]). The embedded com-ponents of ( k ) are not recovered by this technique.In step (v) we control if the irreducible components of V C ( k ) are real, using Algorithm 5.1. omputing real radicals by moment optimization Algorithm 6.1:
Real radical
Input:
Polynomials f = ( 𝑓 , . . . , 𝑓 𝑠 ) ⊂ R [ x ] . 𝑑 : = max ( deg ( f 𝑖 ) , 𝑖 = , . . . , 𝑠 ) − ; success := false;Repeat until success(i) 𝑑 : = 𝑑 + (ii) Compute a generic element 𝜎 ∗ of L 𝑑 + (± f ) (iii) Compute a graded basis k of Ann 𝑑 ( 𝜎 ∗ ) (Algorithm 4.1)(iv) Compute the numerical irreducible components 𝑉 𝑖 of 𝑉 C ( k ) (described by witness sets)(v) For each component 𝑉 𝑖 , check that 𝑉 𝑖 is real(Algorithm 5.1). If not repeat from step (i).(vi) success := true(vii) For each component 𝑉 𝑖 compute defining equations h 𝑖 = { ℎ 𝑖, , . . . , ℎ 𝑖,𝑛 + } of 𝑉 𝑖 Output:
The polynomials h 𝑖 generating the minimal realprime ideals 𝔭 𝑖 lying over ( f ) .In step (vii), the equations defining 𝑉 𝑖 are obtained from 𝑛 + generic projections. In particular, the equation of a generic projec-tion of 𝑉 𝑖 used in step (ii) of Algorithm 5.1 provides one of thedefining equation, say ℎ 𝑖, .We prove the correctness of the algorithm. By Theorem 3.3 wehave V R ( k ) = V R ( f ) for 𝑑 ≥ max ( deg ( f )) . Let 𝔭 𝑖 = ( h 𝑖 ) in step(vii). By construction V R ( k ) = Ð 𝑖 ( 𝑉 𝑖 ) R = Ð 𝑖 V R ( 𝔭 𝑖 ) = V R ( Ñ 𝑖 𝔭 𝑖 ) .If step (v) succeeds, all the 𝔭 𝑖 ’s are real radical, and thus Ñ 𝑖 𝔭 𝑖 isreal radical. Since V R ( f ) = V R ( Ñ 𝑖 𝔭 𝑖 ) , by the Real Nullstellensatz Ñ 𝑖 𝔭 𝑖 = R √ f and the 𝔭 𝑖 are the real prime ideal lying over ( f ) . Theloop stops for some 𝑑 ≫ by Theorem 3.3.Algorithm 6.1 computes the minimal real prime ideals lying over ( f ) , but does not check that the equations k define a real radicalideal. If the ideal ( k ) has no embedded component and the primeideals 𝔭 𝑖 are of multiplicity 1 (checked with the Jacobian criterionfor h at a witness point of 𝔭 𝑖 ), then the success of step (v) impliesthat k = Ann 𝑑 ( 𝜎 ∗ ) defines the real radical of ( f ) .Algorithm 6.1 can be simplified in the case where V R ( f ) is fi-nite. We can check that ( k ) = R √ f , for k = Ann 𝑑 ( 𝜎 ∗ ) , using theflat extension criterion. We can also detect this condition with theinitial of k , see Remark 4.2. In this case, 𝜎 ∗ extends to a positivelinear functional on R [ x ] and ( k ) = R √ f .Similarly, when the ideal ( k ) is prime, one only needs to checkthat it is real (using Algorithm 5.1 on a generic projection), steps(iv), (vii) can be skipped and we obtain ( k ) = R √ f . When ( k ) is realradical, the algorithm can even output directly ( k ) = R √ f . We illustrate Algorithm 6.1, with the Julia package
MomentTools.jl ,using the Semi-Definite Programm optimizer Mosek . Example 7.1.
Let 𝑓 = − 𝑧 + 𝑥 − 𝑥 𝑧 + 𝑥𝑧 + 𝑦𝑧 − 𝑧 − 𝑥 + 𝑥𝑧 − 𝑦 − 𝑧 , 𝑔 = − ( 𝑥 + 𝑦 + 𝑧 ) and 𝑆 = { 𝜉 ∈ R | 𝑓 ( 𝜉 ) = , 𝑔 ( 𝜉 ≥ ) } . We want to compute the 𝑆 -radical of 𝐼 = ( 𝑓 ) , which is equal to ( 𝑧 − 𝑥, 𝑥 − 𝑦 ) . https://gitlab.inria.fr/AlgebraicGeometricModeling/MomentTools.jl X = @polyvar x y zf = -10*z^4 + x^3 - 3*x^2*z + 3*x*z^2 + 20*y*z^2- z^3 - 10*x^2 + 20*x*z - 10*y^2 - 10*z^2g = 5 - (x^2+y^2+z^2)v, M = minimize(one(f),[f], [g], X, 6, optimizer)sigma = get_series(M)[1]L = monomials(X,seq(0:3))K,In,P,B = annihilator(sigma, L)
We compute a generic positive linear functional 𝜎 (by optimisingthe constant function on 𝑆 ), a graded basis K of ( Ann 𝑑 ( 𝜎 )) , theinitial monomials In of K , a basis P of R [ x ]( Ann 𝑑 ( 𝜎 )) orthogonal withrespect to h· , ·i 𝜎 and a monomial basis B of R [ x ]( Ann 𝑑 ( 𝜎 )) . The ele-ments of K are: -0.999999935776211x - 2.027089868945844e-9y + z+ 1.9280308682132505e-9x ² - 1.9114608711668615e-8x - 0.9999998601127081y- 2.6012502193917264e-7 These polynomials define a parametrisation of parabola and thusgenerate a real radical ideal. They are approximation of the gener-ators of the 𝑆 -radical of 𝐼 within an error . Example 7.2.
We compute equations for the hold of the Whitneyumbrella. Let 𝑓 = 𝑥 − 𝑦 𝑧, 𝑔 = − ( 𝑥 + 𝑦 + ( 𝑧 + ) ) and 𝑆 = { 𝜉 ∈ R | 𝑓 ( 𝜉 ) = , 𝑔 ( 𝜉 ≥ ) } . We compute the 𝑆 -radical of 𝐼 = ( 𝑓 ) , which is equal to ( 𝑥, 𝑦 ) . Proceding as above, we obtain for K , the polynomials: x + 3.1388489268444904e-21y + 3.6567022687420305e-21 These polynomials are a good approximation of the generators ( 𝑥, 𝑦 ) of the real radical, defining the singular locus of the Whit-ney umbrella. Example 7.3.
This example is taken from [30, ex. 9.6]. We wantto compute the real radical of 𝐼 = ( 𝑓 , 𝑓 , 𝑓 ) ⊂ R [ 𝑥, 𝑦, 𝑧 ] , where: 𝑓 = 𝑥 + 𝑥 𝑦 − 𝑥 𝑧 − 𝑥 − 𝑦 + 𝑧𝑓 = 𝑥 𝑦 + 𝑦 − 𝑦 𝑧 − 𝑥 − 𝑦 + 𝑧𝑓 = 𝑥 𝑧 + 𝑦 𝑧 − 𝑧 − 𝑥 − 𝑦 + 𝑧. Its variety has three irreducible components, two lines and a point,defined by the real prime ideals 𝔭 = ( 𝑥 − 𝑧, 𝑦 ) , 𝔭 = ( 𝑥 − 𝑧 + , 𝑦 − ) and 𝔪 = ( 𝑥 − , 𝑦 − , 𝑧 − ) . In the primary decomposition of 𝐼 there is an embedded component 𝔪 ′ , corresponding to the point ( , , ) ∈ V ( 𝔭 ) which has multiplicity two. The real radical of 𝐼 is R √ 𝐼 = 𝔭 ∩ 𝔭 ∩ 𝔪 = ( 𝑦 − 𝑦, 𝑥 − 𝑥𝑧 + 𝑧 + 𝑥 − 𝑧, 𝑥𝑧 + 𝑦𝑧 − 𝑧 − 𝑥 − 𝑦 + 𝑧, 𝑥𝑦 + 𝑥𝑧 − 𝑧 − 𝑥 − 𝑦 + 𝑧 ) .We compute R √ 𝐼 as described in the algorithm. v, M = minimize(one(f1),[f1,f2,f3], [], X, 8, optimizer)sigma = get_series(M)[1]L = monomials(X,seq(0:3))K,I,P,B = annihilator(sigma, L) The elements of K are: -0.9999999985579915x ² - 0.9999999940764733xy + xz +0.9999999838152133x + 0.9999999868597321y -0.9999999838041349z - 2.550976860304921e-104.386341684978274e-7x ² + 3.2135911001749273e-7xy + y ² orenzo Baldi & Bernard Mourrain - 8.511512801700947e-7x - 1.0000008530709377y+ 9.888494964176088e-7z - 5.851033908621897e-88.763853490689755e-7x ² - 0.9999993625797754xy + yz +0.9999983122334805x - 1.6948939787209127e-6y -0.9999980367703514z - 1.1680315895740145e-7-0.9999991215344914x ² - 1.99999935020258xy + z ² +2.99999828318184x + 1.9999982828997438y -2.999998007995895z - 1.1724998920381591e-7 which are approximately (within an error of ) generatorsof R √ 𝐼 . Example 7.4.
This example is taken from [11, 8.2]. We want tocompute the real radical of 𝐼 = ( 𝑓 , 𝑓 , 𝑓 ) ⊂ R [ 𝑥, 𝑦, 𝑧 ] , where: 𝑓 = 𝑥𝑦𝑧𝑓 = 𝑧 ( 𝑥 + 𝑦 + 𝑧 + 𝑦 ) 𝑓 = 𝑦 ( 𝑦 + 𝑧 ) . The associated complex variety has four irreducible components:two conjugates lines intersecting in the origin, another line (doublefor f ) and a point. The real variety is given by a line 𝔭 = ( 𝑦, 𝑧 ) anda point 𝔪 = ( 𝑥, 𝑦 + , 𝑥 − ) . The real radical is R √ 𝐼 = 𝔭 ∩ 𝔪 = ( 𝑦𝑥, 𝑧 + 𝑦, 𝑦 + 𝑦 ) . A direct check shows that these polynomialsgenerate a real radical ideal.We compute R √ 𝐼 as described above and obtain for K : -6.53338688785662e-19x + 0.9995827809845268y + z- 0.00020850768649473272-1.4685109255649737e-19x ² + xy + 5.9730164512226755e-6x+ 2.1320912413237275e-19y + 1.0655056374451632e-19-2.268705086623265e-6x ² + y ² + 1.88498770272315e-19x +0.4998194337295852y + 4.384653173789382e-6 approximating (within an error of ) the generators of R √ 𝐼 . Algorithm 6.1 is a symbolic-numeric algorithm, which output de-pends on the quality of the numerical tools that are involved. Inparticular, the numerical quality of the generic positive linear func-tional 𝜎 ∗ , produced by a SDP solver, impacts the computation ofgenerators of the real radical. This computation depends on a thresh-old used to determine when a polynomial is in annihilator. A de-tailled analysis of the numerics behind the algorithm as well asbounds analysing its complexity are left for futur investigations. REFERENCES [1] Michael Artin. 2017.
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