Computing rational points in convex semi-algebraic sets and SOS decompositions
Let ${\cal P}=\{h_1, ..., h_s\}\subset \Z[Y_1, ..., Y_k]$, $D\geq \deg(h_i)$ for $1\leq i \leq s$, $\sigma$ bounding the bit length of the coefficients of the $h_i$'s, and $\Phi$ be a quantifier-free ${\cal P}$-formula defining a convex semi-algebrai…
Authors: Mohab Safey El Din (LIP6, INRIA Rocquencourt), Lihong Zhi (KLMM)
apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7045--FR+ENG INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Computing rational points in con v ex semi-alge braic sets and SOS decompositions Mohab Safe y El Di n — Lihong Zhi N° 7045 Septembre 2009 Centre de recher che INRIA Paris – Rocquencour t Domaine de V oluceau, Rocquen court, BP 105, 78153 Le Chesnay Cedex Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30 Computing rational p oin ts in on v ex semi-algebrai sets and SOS deomp ositions Mohab Safey El Din ∗ , Lihong Zhi † Thème : Algorithmique, alul ertié et ryptographie Équip e-Pro jet SALSA Rapp ort de re her he n ° 7045 Septem bre 2009 18 pages Abstrat: Let P = { h 1 , . . . , h s } ⊂ Z [ Y 1 , . . . , Y k ] , D ≥ deg( h i ) for 1 ≤ i ≤ s , σ b ounding the bit length of the o eien ts of the h i 's, and Φ b e a quan tier-free P -form ula dening a on v ex semi-algebrai set. W e design an algorithm return- ing a rational p oin t in S if and only if S ∩ Q 6 = ∅ . It requires σ O(1) D O( k 3 ) bit op erations. If a rational p oin t is outputted its o ordinates ha v e bit length dom- inated b y σ D O( k 3 ) . Using this result, w e obtain a pro edure deiding if a p oly- nomial f ∈ Z [ X 1 , . . . , X n ] is a sum of squares of p olynomials in Q [ X 1 , . . . , X n ] . Denote b y d the degree of f , τ the maxim um bit length of the o eien ts in f , D = n + d n and k ≤ D ( D + 1) − n +2 d n . This pro edure requires τ O(1) D O( k 3 ) bit op erations and the o eien ts of the outputted p olynomials ha v e bit length dominated b y τ D O( k 3 ) . Key-w ords: rational sum of squares, semidenite programming, on v ex semi- algebrai sets, omplexit y . ∗ UPMC, P aris 6, LIP6 † KLMM, A adem y of Mathematis and System Sienes, China Calul de p oin ts rationnels dans des semi-algébriques on v exes et déomp osition en sommes de arrés Résumé : Soit P = { h 1 , . . . , h s } ⊂ Z [ Y 1 , . . . , Y k ] , D ≥ deg( h i ) p our 1 ≤ i ≤ s , σ une b orne sur la longueur binaire des o eien ts des h i , et Φ une P - form ule sans quan tiateurs dénissan t un ensem ble semi-algébrique on v exe. Nous dériv ons un algorithme qui retourne un p oin t à o ordonnées rationnelles dans S si et seulemen t si S ∩ Q 6 = ∅ . Cet algorithme est de omplexité binaire σ O(1) D O( k 3 ) . Si un p oin t rationnel est ren v o y é, ses o ordonnées son t de longueur binaires dominées par σ D O( k 3 ) . On déduit de e résultat une pro édure qui déide si un p olynme f ∈ Z [ X 1 , . . . , X n ] est une somme de arrés de p olynmes dans Q [ X 1 , . . . , X n ] . Soit d le degré de f , τ le maxim um des longueurs binaires des o eien ts de f , D = n + d n et k ≤ D ( D + 1) − n +2 d n . Cette pro édure est de omplexité binaire τ O(1) D O( k 3 ) et les o eien ts des p olynmes obten us en sortie on t une longueur binaire dominée par τ D O( k 3 ) . Mots-lés : sommes de arrés à o eien ts rationnels, programmation semi- dénie p ositiv e, ensem bles semi-algebraiques on v exes, omplexité. R ational p oints in onvex semi-algebr ai sets 3 1 In tro dution Motiv ation and problem statemen t. Supp ose f ∈ R [ x 1 , . . . , x n ] , then f is a sum of squares (SOS) in R [ x 1 , . . . , x n ] if and only if it an b e written in the form f = v T · M · v , (1) in whi h v is a olumn v etor of monomials and M is a real p ositiv e semidenite matrix (P o w ers and Wörmann , 1998 , Theorem 1) (see also Choi et al. (1995 )). M is also alled a Gr am matrix for f . If M has rational en tries, then f is a sum of squares in Q [ x 1 , . . . , x n ] . PR OBLEM 1.1 (Sturmfels). If f ∈ Q [ x 1 , . . . , x n ] is a sum of squar es in R [ x 1 , . . . , x n ] , then is f also a sum of squar es in Q [ x 1 , . . . , x n ] ? It has b een p oin ted out that if there is an in v ertible Gram matrix for f , then there is a Gram matrix for f with rational en tries (Hillar, 2009 , Theorem 1.2). F urthermore, if f ∈ Q [ x 1 , . . . , x n ] is a sum of m squares in K [ x 1 , . . . , x n ] , where K is a totally real n um b er eld with Galois losure L , then f is also a sum of 4 m · 2 [ L : Q ]+1 [ L : Q ]+1 2 squares in Q [ x 1 , . . . , x n ] (Hillar , 2009 , Theorem 1.4). It is in teresting to see that the n um b er of squares an b e redued to m (see Kaltofen (2009 )). Although no example is kno wn of a rational p olynomial ha ving only irra- tional sum of squares, a omplete answ er to Question 1.1 is not kno wn. This is the main motiv ation for us to design an algorithm to he k whether a rational p olynomial ha ving a rational sum of squares deomp osition and giv e the rational SOS represen tation if it do es exist. By reduing this problem to semi-denite programming, this an b e done b y designing an algorithm he king if a on v ex semi-algebrai set on tains rational p oin ts (see P o w ers and Wörmann (1998 )). Main result. W e prop ose an algorithm whi h deides if a onvex semi-algebrai set S ⊂ R k on tains rational p oin ts (i.e. p oin ts with o ordinates in Q k ). In the ase where S ∩ Q k is non-empt y , a rational p oin t in S is omputed. The semi-algebrai set S is giv en as the solution set of a p olynomial system of non-strit inequalities with in teger o eien ts. Arithmeti op erations, sign ev aluations and omparisons of t w o in tegers/rationals an b e done in p olynomial time of the maxim um bit length of the onsidered in tegers/rationals. W e b ound the n um b er of bit op erations that the algorithm p erforms with resp et to the n um b er of p olynomials, their degrees and the maxim um bit length of their input o eien ts; w e also giv e upp er b ounds on the bit length of the o- ordinates of the outputted rational p oin t if this situation o urs. More preisely , the main result is as follo ws. Theorem 1.1 Consider a set of p olynomials P = { h 1 , . . . , h s } ⊂ Z [ Y 1 , . . . , Y k ] , and a quantier-fr e e P -formula Φ( Y 1 , . . . , Y k ) and let D b e an inte ger suh that deg( h i ) ≤ D for 1 ≤ i ≤ s and σ the maximum bit length of the o eients of the h i 's. L et S ⊂ R k b e the onvex semi-algebr ai set dene d by Φ . Ther e exists an algorithm whih de ides if S ∩ Q k is non-empty within σ O(1) ( sD ) O( k 3 ) bit op er ations. In ase of non-emptiness, it r eturns an element of S ∩ Q k whose o or dinates have bit length dominate d by σ D O( k 3 ) . RR n ° 7045 4 Safey El Din & Zhi W e use a pro edure due to Basu et al. (1996 ) p erforming quan tier elimina- tion o v er the reals in order to dedue from Theorem 1.1 the follo wing result. Corollary 1.2 L et S ⊂ R k b e a onvex set dene d by S = { Y ∈ R k : ( Q 1 X [1] ∈ R n 1 ) · · · ( Q ω X [ ω ] ∈ R n ω ) P ( Y , X [1] , . . . , X [ ω ] ) } with quantiers Q i ∈ {∃ , ∀} , wher e X [ i ] is a set of n i variables, P is a Bo ole an funtion of s atomi pr e di ates g ( Y , X [1] , . . . , X [ ω ] ) ∆ i 0 wher e ∆ i ∈ { >, < , = } (for i = 1 , . . . , s ) and the g i 's ar e p olynomials of de gr e e D with inte ger o eients of binary size at most σ . Ther e exists an algorithm whih de ides if S ∩ Q k is non-empty within σ O(1) ( sD ) O( k 3 Π ω i =1 n i ) bit op er ations. In ase of non-emptiness, it r eturns an element of S ∩ Q k whose o or dinates have bit length dominate d by σ D O( k 3 Π ω i =1 n i ) . The pro of of the ab o v e results is based on quan titativ e and algorithmi re- sults for omputing sampling p oin ts in semi-algebrai sets and quan tier elimi- nation o v er the reals. It is w ell-kno wn that deiding if a giv en p olynomial f ∈ Z [ X 1 , . . . , X n ] of degree d whose o eien ts ha v e bit length dominated b y τ is a sum of squares of p olynomials in Q [ X 1 , . . . , X n ] an b e redued to a linear matrix inequalit y whi h denes a on v ex semi-algebrai set (see e.g. P o w ers and Wörmann (1998 )). Applying Theorem 1.1 , w e sho w that there exists an algorithm deiding if su h an SOS deomp osition exists o v er the rationals and that the o eien ts of the p olynomials in the deomp osition ha v e bit length dominated b y τ D O( k 3 ) with D = n + d n and k ≤ D ( D + 1) − n +2 d n . Moreo v er, su h a deomp osition an b e found within τ O(1) D O( k 3 ) bit op erations. Prior w orks. Kha hiy an and P ork olab extended the w ell-kno wn result of Lenstra (1983 ) on the p olynomial-time solv abilit y of linear in teger program- ming in xed dimension to semidenite in teger programming. The follo wing prop osition is giv en in Kha hiy an and P ork olab (1997 , 2000 ). Prop osition 1.3 L et S ⊂ R k b e a onvex set dene d as in Cor ol lary 1.2 . Ther e exists an algorithm for solving the pr oblem min { Y k | Y = ( Y 1 , . . . , Y k ) ∈ S T Z k } in time ℓ O(1) ( sD ) O( k 4 )Π ω i =1 O( n i ) . In ase of non-empty, then the min- imization pr oblem has an optimal solution whose bit length is dominate d by ℓD O( k 4 )Π ω i =1 O( n i ) . Their algorithm w as further impro v ed b y Heinz for the ase of on v ex min- imization where the feasible region is desrib ed b y quasion v ex p olynomials Heinz (2005 ). Although w e an apply Prop osition 1.3 diretly to ertify that a giv en p oly- nomial with in teger o eien ts to b e non-negativ e for all real v alues of the v ariables b y omputing a sum of squares in Z [ x 1 , . . . , x n ] , the nonnegativit y of a p olynomial an b e ertied if it an b e written as a sum of squares of p olynomials in Q [ x 1 , . . . , x n ] . Some h ybrid sym b oli-n umeri algorithms ha v e b een giv en in P eyrl and P arrilo (2007 , 2008 ); Kaltofen et al. (2008 , 2009 ) whi h INRIA R ational p oints in onvex semi-algebr ai sets 5 turn a n umerial sum of squares represen tation of a p ositiv e p olynomial in to an exat rational iden tit y . Ho w ev er, it is w ell kno wn that there are plen t y of p olynomials whi h are nonnegativ e but an not b e written as sums of squares of p olynomials, for example, the famous Motzkin p olynomial. This also imp el us to study Kha hiy an and P ork olab's approa h. It turns out that b y fo using on rational n um b ers instead of in tegers, w e an design an exat algorithm whi h deide whether a giv en p olynomial an b e written as an SOS o v er the rationals and giv e the rational SOS deomp osition if it exists. Struture of the pap er. Setion 2 is dev oted to reall the quan titativ e and algorithmi results on omputing sampling p oin ts in semi-algebrai sets and quan tier elimination o v er the reals. Most of these results are pro v ed in Basu et al. (1996 ). Setion 3 is dev oted to pro v e the orretness of the algo- rithm on whi h Theorem 1.1 and Corollary 1.2 rely . The omplexit y analysis is done in Setion 4 . In Setion 5, w e apply Theorem 1.1 to pro v e the announed b ounds on the bit length of the rational o eien ts of the deomp osition in to sums of squares of a giv en p olynomial with in teger o eien ts. A kn wledgmen ts. This w ork is supp orted b y the EXA CT A gran t of National Siene F oundation of China (NSF C) and the F ren h National Resear h Ageny (ANR). The authors thank INRIA, KLMM and the A adem y of Mathematis and System Sienes for their supp ort. 2 Preliminaries The algorithm on whi h Theorem 1.1 relies and its omplexit y analysis are based on algorithmi and quan titativ e results on omputing sampling p oin ts in semi-algebrai sets and quan tier elimination o v er the reals. 2.1 Computing p oin ts in semi-algebrai sets Consider a set of p olynomials P = { h 1 , . . . , h J } ⊂ Z [ Y 1 , . . . , Y k ] , and a quan tier- free P -form ula Φ( Y 1 , . . . , Y k ) (i.e. a quan tier-free form ula whose atoms is one of h = 0 , h 6 = 0 , h > 0 , h < 0 for h ∈ P ). Let D b e an in teger su h that deg( h i ) ≤ D for 1 ≤ i ≤ J and ℓ the maxim um bit length of the o eien ts of the h i 's. W e denote b y S ⊂ R k the semi-algebrai set dened b y Φ( Y 1 , . . . , Y k ) . A funtion RealizableSignConditions omputing a set of algebrai p oin ts ha v- ing a non-empt y in tersetion with ea h onneted omp onen t of semi-algebrai sets dened b y sign onditions satised b y P is giv en in ( Basu et al. , 1996 , Setion 3) (see also (Basu et al. , 2006 , Chapter 5)). From this, a funtion Sam- plingP oints omputing a set of algebrai p oin ts ha ving a non-empt y in tersetion with ea h onneted omp onen t of S is obtained. These algebrai p oin ts are eno ded b y a rational parametrization G = 0 , Y 1 = G 1 G 0 , . . . , Y k = G k G 0 RR n ° 7045 6 Safey El Din & Zhi where G, G 0 , . . . , G k are p olynomials in Z [ T ] su h that deg(gcd( G, G 0 )) = 0 and for 1 ≤ i ≤ k , − 1 ≤ deg( G i ) ≤ deg( G ) − 1 and 0 ≤ deg ( G 0 ) ≤ deg ( G ) − 1; the rational parametrization is giv en b y the list G = ( G, G 0 , G 1 , . . . , G k ) ; the degree of G is alled de gr e e of the r ational p ar ametrization and Z ( G ) ⊂ C k denotes the set of omplex p oin ts eno ded b y G ; and a list T of the Thom-eno dings of the real ro ots ϑ of G su h that Φ G 1 ( ϑ ) G 0 ( ϑ ) , . . . , G k ( ϑ ) G 0 ( ϑ ) is true. The bit omplexit y of SamplingP oints is ℓJ k +1 D O( k ) and the output is su h that deg( G ) = O( D ) k and the bit length of the o eien ts of G, G 0 , G 1 , . . . , G k is dominated b y ℓD O( k ) . F atorizing o v er Q a univ ariate p olynomial h ∈ Q [ T ] of degree δ with rational o eien ts of maxim um bit length ℓ an b e done in ℓ O(1) δ O(1) bit-op erations (see Lenstra et al. (1982 ); v an Ho eij and No v o in (2007 ); S hönhage (1984 )). Giv en a ro ot ϑ of h , the minimal p olynomial of ϑ has o eien ts of bit length dominated b y ℓ + O( δ ) (see Mignotte (1982 )). Consider no w a ro ot ϑ of G and its minimal p olynomial g . Sine G and G 0 are o-prime, one an ompute G − 1 0 mo d g to obtain a rational parametrization ( g , g 0 , . . . , g k ) with in teger o eien ts of bit length dominated b y ℓD O( k ) and for 1 ≤ i ≤ k , − 1 ≤ deg ( g i ) ≤ deg ( g ) − 1 and 0 ≤ deg ( g 0 ) ≤ deg ( g ) − 1 within a bit-omplexit y ℓ O(1) D O( k ) . This implies the follo wing result. Prop osition 2.1 Ther e exists a funtion SemiAlgeb raiSolve whih takes as input the system Φ( Y 1 , . . . , Y k ) and omputes a r ational p ar ametrization G = ( G, G 0 , G 1 , . . . , G k ) and a list T of Thom-en o dings suh that G is irr e duible over Q , and T ontains the en o dings of the r e al r o ots ϑ of G suh that G 1 ( ϑ ) G 0 ( ϑ ) , . . . , G k ( ϑ ) G 0 ( ϑ ) ∈ S . The bit length of the o eients of G, G 0 , G 1 , . . . , G k is dominate d by ℓD O( k ) and deg( G ) = O( D ) k . Mor e over, SemiAlgeb raiSolve r e quir es ℓ O(1) J k +1 D O( k ) bit op er ations. Remark 2.2 Sin e G and G 0 ar e o-prime, one an ompute G 0 − 1 mo d G in p olynomial time, and the binary length of its r ational o eients an b e b ounde d via subr esultants, we an assume, without loss of gener ality, that the r ational p ar ametrization has a onstant denominator: Y = 1 q ( G 1 ( ϑ ) , G 2 ( ϑ ) , . . . , G k ( ϑ )) ∈ S , G ( ϑ ) = 0 , (2) wher e the bit length of q and the o eients of G, G 1 , . . . , G k ar e dominate d by ℓD O( k ) . The ab o v e disussion leads also to the follo wing result. Prop osition 2.3 L et G , T b e the output of SemiAlgebraicSolve (Φ) , δ b e the de gr e e of G , and ℓ b e the maximum bit length of the o eients of the p oly- nomials in G ∪ P . Ther e exists a funtion RationalZeroDimSolve whih takes INRIA R ational p oints in onvex semi-algebr ai sets 7 as input G and Φ and r eturns a r ational p oint y ∈ Z ( G ) if and only if y ∈ S ∩ Z ( G ) ∩ Q k , else it r eturns an empty list. The o or dinates of these r atio- nal p oints have bit length dominate d by ℓδ O(1) and omputations ar e p erforme d within O( k )O( J ) ℓ O(1) δ O(1) n + D n O(1) bit op er ations. Remark 2.4 A or ding to Pr op osition 2.1 , the funtion SemiAlgeb raiSolve om- putes a r ational p ar ametrization G = ( G, G 0 , G 1 , . . . , G k ) suh that G is irr e- duible over Q . Ther efor e a r ational p oint y ∈ Z ( G ) if and only if deg( G ) = 1 . In or der to he k whether y ∈ S , we only ne e d to evaluate the formula Φ at y . The follo wing result is a restatemen t of (Basu et al. , 1996 , Theorem 4.1.2) and allo ws us to b ound the bit length of rational p oin ts in non-empt y semi- algebrai sets dened b y strit p olynomial inequalities. Prop osition 2.5 L et S ′ ⊂ R k b e a semi-algebr ai set dene d by a quantier- fr e e P -formula whose atoms ar e strit ine qualities. Then S ′ ontains a r ational p oint whose o or dinates have bit length dominate d by ℓD O ( k ) . The pro of of the ab o v e result (see (Basu et al., 1996 , Pro of of Theorem 4.1.2 pp. 1032)) is based on the routine RealizableSignConditions and the iso- lation of real ro ots of univ ariate p olynomials with rational o eien ts (see e.g. (Basu et al. , 2006 , Chapter 10)). W e denote b y RationalOp enSemiAlgeb raiSolve a funtion taking as input the P -form ula Φ and whi h returns a rational p oin t in S if and only if there exists a non-empt y semi-algebrai set S ′ dened b y a quan tier-free P -form ula whose atoms are strit inequalities su h that S ′ ⊂ S . The result b elo w is not stated in Basu et al. (1996 ) but is an immediate onse- quene of this pro of. Corollary 2.6 Supp ose that ther e exists a quantier-fr e e P -formula whose atoms ar e strit ine qualities dening a non-empty semi-algebr ai set S ′ ⊂ S . Ther e exists an algorithm omputing a r ational p oint in S if and only if S 6 = ∅ . It r e quir es ℓ O(1) J k +1 D O( k ) bit op er ations and if a r ational p oint is outputte d, its o or dinates have bit length dominate d by ℓD O( k ) . 2.2 Quan tier elimination o v er the reals W e onsider no w a rst-order form ula F o v er the reals ( Q 1 X [1] ∈ R n 1 ) · · · ( Q ω X [ ω ] ∈ R n ω ) P ( Y , X [1] , . . . , X [ ω ] ) where Y = ( Y 1 , . . . , Y k ) is the v etor of free v ariables; ea h Q i ( i = 1 , . . . , ω ) is one of the quan tiers ∃ or ∀ ; P ( Y , X [1] , . . . , X [ ω ] ) is a Bo olean funtion of s atomi prediates g ( Y , X [1] , . . . , X [ ω ] ) ∆ i 0 where ∆ i ∈ { >, <, = } (for i = 1 , . . . , s ) and the g i 's are p olynomials of degree D with in teger o eien ts of binary size at most ℓ . RR n ° 7045 8 Safey El Din & Zhi The follo wing result on quan tier elimination is a restatemen t of (Basu et al., 1996 , Theorem 1.3.1). Theorem 2.7 Ther e exists a quantie d-fr e e formula Ψ I _ i =1 J i ^ j =1 ( h ij ∆ ij 0) (wher e h ij ∈ Z [ Y 1 , . . . , Y k ] and ∆ ij ∈ { = , > } ) whih is e quivalent to F and suh that I ≤ s ( k +1)Π ω i =1 ( n i +1) D ( k +1)Π ω i =1 O( n i ) , J i ≤ s Π ω i =1 ( n i +1) D Π ω i =1 O( n i ) , deg( h ij ) ≤ D Π ω i =1 O( n i ) , the bit length of the o eients of the p olynomials h ij is dominate d by ℓD ( k +1)Π ω i =1 O( n i ) . The ab ove tr ansformation r e quir es ℓs ( k +1)Π ω i =1 ( n i +1) D ( k +1)Π ω i =1 O( n i ) bit op er a- tions. In the sequel, w e denote b y QuantierElimination a funtion that tak es F as input and returns a list [Ψ 1 , . . . , Ψ I ] where the Ψ ′ i s are the onjuntions J i ^ j =1 ( h ij ∆ ij 0) . 3 Algorithm and orretness 3.1 Desription of the algorithm W e use the follo wing funtions: Substitute whi h tak es as input a v ariable Y r ∈ { Y 1 , . . . , Y k } , a p olynomial h ∈ Q [ Y 1 , . . . , Y k ] and a Bo olean form ula F and whi h returns a form ula ˜ F obtained b y substituting Y r b y h in F . RemoveDenominato rs whi h tak es as input a form ula F and returns a for- m ula ˜ F obtained b y m ultiplying the p olynomials in F b y the absolute v alue of the lm of the denominators of their o eien ts. Consider no w a rational parametrization G = ( G, G 0 , G 1 , . . . , G k , G k +1 ) ⊂ Z [ T ] k +3 with δ = deg ( G ) . F or 0 ≤ i ≤ δ − 1 , denote b y a i ∈ Z k the v etor of in tegers whose j -th o ordinate is the o eien t of T i in G j . Similarly , for 0 ≤ i ≤ δ − 1 , b i denotes the o eien t of T i in G k +1 . W e use in the sequel a funtion Generate V eto rs that tak es as input a rational parametrization G . This funtion returns the set list of ouples ( a i , b i ) for 0 ≤ i ≤ δ − 1 . As in the previous setion, onsider no w a set of p olynomials P = { h 1 , . . . , h s } ⊂ Z [ Y 1 , . . . , Y k ] , and a quan tier-free P -form ula Φ( Y 1 , . . . , Y k ) and let D b e an in- teger su h that deg( h i ) ≤ D for 1 ≤ i ≤ s and σ the maxim um bit length of the INRIA R ational p oints in onvex semi-algebr ai sets 9 o eien ts of the h i 's. W e denote b y S ⊂ R k the semi-algebrai set dened b y Φ( Y 1 , . . . , Y k ) whi h is supp osed to b e on v ex. The routine FindRationalP oints b elo w tak es as input the form ula Φ( Y 1 , . . . , Y k ) dening S ⊂ R k and the list of v ariables [ Y 1 , . . . , Y k ] . FindRationalP oints ( Φ , [ Y 1 , . . . , Y k ] ). 1. Let L = Ratio nalOp enSemiAlgebraicSolve ( Open (Φ)) 2. If L is not empt y then return L 3. Let G , T = SemiAlgebraicSolve (Φ) 4. If T is empt y then return [℄ 5. Let L = Ratio nalZeroDimSol ve ( G , Φ) 6. If L is not empt y or k = 1 then return L 7. Else (a) Let A 1 , . . . , A k , B b e free v ariables and Θ b e the form ula ∀ Y ∈ R k A 2 1 + · · · + A 2 k > 0 ∧ ( ¬ Φ ∨ A 1 Y 1 + · · · + A k Y k = B ) (b) Let [Ψ 1 , . . . , Ψ I ] = Quantifi erEliminatio n (Θ) and i = 1 () While i ≤ I do i. G , T = SemiAlgebraicSolve ( Ψ i ) and ( G, G 0 , G 1 , . . . , G k , G k +1 ) = G ii. If T is empt y i = i + 1 else break. (d) Let C = GenerateVector s ( G, G 0 , G 1 , . . . , G k , G k +1 ) (e) Let a = ( a 1 , . . . , a k ) 6 = (0 , . . . , 0) and b ∈ Z su h that ( a, b ) ∈ C (f ) Let r = max( i, 1 ≤ i ≤ k and a i 6 = 0) (g) Let h = b − P r − 1 j =1 a i Y i a r (h) Let Φ ′ = RemoveDe n ominators ( Substitute ( Y r , h, Φ)) (i) Let L = FindRatio nalPoints (Φ ′ , [ Y 1 , . . . , Y r − 1 , Y r +1 , . . . , Y k ]) (j) If L is not empt y , i. Let ( q 1 , . . . , q r − 1 , q r +1 , . . . , q k ) b e its elemen t; ii. Let q r = Evaluate ( { Y i = q i , 1 ≤ i ≤ k , j 6 = r } , h ) iii. if Φ( q 1 , . . . , q r − 1 , q r , q r +1 , . . . , q k ) is true, return [( q 1 , . . . , q r − 1 , q r , q r +1 , . . . , q k )] else return [℄. (k) Else return [℄. Prop osition 3.1 The algorithm FindRationalP oints r eturns a list ontaining a r ational p oint if and only if S ∩ Q k is non-empty, else it r eturns an empty list. The next paragraph is dev oted to pro v e this prop osition. RR n ° 7045 10 Safey El Din & Zhi Remark 3.2 L et S ⊂ R k b e a onvex set dene d by S = { Y ∈ R k : R k ( Q 1 X [1] ∈ R n 1 ) · · · ( Q ω X [ ω ] ∈ R n ω ) P ( Y , X [1] , . . . , X [ ω ] ) } with quantiers Q i ∈ {∃ , ∀} , wher e X [ i ] is a set of n i variables, P is a Bo ole an funtion of s atomi pr e di ates g ( Y , X [1] , . . . , X [ ω ] ) ∆ i 0 wher e ∆ i ∈ { >, <, = } (for i = 1 , . . . , s ). Denote by Θ the quantie d formula dening S and by [Ψ 1 , . . . , Ψ I ] the output of QuantifierElimi nation (Θ) . R unning FindRationalP oints on the Ψ i 's al lows to de ide the existen e of r ational p oints in S . This pr oves a p art of Cor ol lary 1.2 . 3.2 Pro of of orretness In the sequel, w e denote b y clos Zar ( S ) its Zariski-losure. F ollo wing (Bo hnak et al. , 1998 , Denition 2.8.1 and Prop osition 2.8.2 pp. 50), w e dene the dimension of S as the Krull dimension of the ideal asso iated to clos Zar ( S ) . By on v en tion, the dimension of the empt y set is − 1 . W e reuse the notations in tro dued in the desription of FindRationalP oints . The pro of is done b y indution on k . Before in v estigating the ase k = 1 , w e reall some elemen tary fats. Preliminaries. W e start with a lemma. Lemma 3.3 L et A ⊂ R k b e a semi-algebr ai set dene d by a quantier-fr e e P -formula. If dim( A ) = k ther e exists y ∈ R k suh that for al l h ∈ P h ( y ) > 0 or h ( y ) < 0 . Pr o of. Supp ose that for all y ∈ A , there exists h ∈ P su h that h ( y ) = 0 . Then, A is on tained in the union H of the h yp ersurfaes dened b y h = 0 for h ∈ P . Consequen tly , dim( A ) ≤ dim( H ) < k , whi h on tradits dim( A ) = k . The follo wing lemma realls an elemen tary prop ert y of on v ex semi-algebrai sets of dimension 0 . Lemma 3.4 L et A ⊂ R k b e a onvex semi-algebr ai set. If dim( A ) = 0 , then A is r e du e d to a single p oint. Pr o of. If there exist t w o distint p oin ts y 1 , y 2 in A , the set B = { ty 1 + (1 − t ) y 2 , t ∈ [0 , 1] } is on tained in A . This implies that clos Zar ( B ) ⊂ clos Zar ( A ) and onsequen tly dim( B ) ≤ dim( A ) . Sine clos Zar ( B ) is the line on taining y 1 and y 2 , dim( B ) = 1 and dim( A ) ≥ 1 whi h on tradits the assumption dim( A ) = 0 . Our laim follo ws. Corretness when k = 1 . Lemma 3.5 Supp ose that k = 1 . Then Steps ( 1 -6) r eturn a r ational p oint in S if and only if S ∩ Q k 6 = ∅ else an empty list is r eturne d. Pr o of. If k = 1 , the dimension of S is either 1 , − 1 or 0 . INRIA R ational p oints in onvex semi-algebr ai sets 11 1. Supp ose that S has dimension 1 . From Lemma 3.3 , there exists a non- empt y semi-algebrai set S ′ ⊂ S dened b y a quan tier-free P -form ula whose atoms are strit inequalities. Th us S ′ on tains a rational p oin t. From Corollary 2.6 , su h a rational p oin t in S is outputted at Step ( 1). 2. Supp ose that S has dimension − 1 (i.e. S is empt y). From Prop osition 2.1 , the list of Thom-eno dings outputted at Step (3) is empt y and the empt y list is returned at Step (4). 3. Supp ose that S has dimension 0 . From Lemma 3.4 , S is a single p oin t on tained in Z ( G ) . From Prop osition 2.3, this p oin t is outputted at Step (5 ) if and only if it is a rational p oin t; else the empt y list is outputted. The ase k > 1 . Our indution assumption is that, giv en a quan tier-free P ′ -form ula Φ ′ (with P ′ ⊂ Z [ Y 1 , . . . , Y k − 1 ] ) dening a on v ex semi-algebrai set S ′ ⊂ R k − 1 , FindRa- tionalP oints returns a list on taining a rational p oin t if and only if S ′ ∩ Q k − 1 is non-empt y , else it returns an empt y list. Lemma 3.6 Supp ose that 0 ≤ dim( S ) < k . Ther e exists ( a 1 , . . . , a k ) ∈ R k and b ∈ R suh that ( a 1 , . . . , a k ) 6 = (0 , . . . , 0) and ∀ ( y 1 , . . . , y k ) ∈ R k ( y 1 , . . . , y k ) ∈ S = ⇒ a 1 y 1 + · · · + a k y k = b. (3) Pr o of. It is suien t to pro v e that clos Zar ( S ) is an ane subspae o v er R : in this ase, there exists a real ane h yp erplane H (dened b y P k i =1 a i Y i = b for ( a 1 , . . . , a k ) ∈ R k \ (0 , . . . , 0) and b ∈ R ) su h that S ⊂ clo s Zar ( S ) ⊂ H . W e pro v e b elo w that clos Zar ( S ) ∩ R k is an ane subspae whi h implies that clos Zar ( S ) is an ane subspae. From Lemma 3.4 , if dim( S ) = 0 then S is a single p oin t; th us the onlusion follo ws immediately . W e supp ose no w that dim( S ) > 0 ; hene S is not empt y and on tains in- nitely man y p oin ts. Consider y 0 ∈ S . Giv en y ∈ R k \ { y 0 } , w e denote b y L y 0 ,y ⊂ R k the real line on taining y and y 0 and b y H y 0 ,y ⊂ R k the real ane h yp erplane whi h is orthogonal to L y 0 ,y and whi h on tains y 0 . Sine S is on v ex, for all y ∈ S \{ y 0 } , S ∩ L y ,y 0 6 = ∅ . W e onsider the set U y 0 = T y ∈S \{ y 0 } H y 0 ,y ; note that U y 0 is an ane subspae sine it is the in tersetion of ane subspaes. W e laim that the orthogonal of U y 0 is clos Zar ( S ) ∩ R k . W e rst pro v e that S is on tained in the orthogonal of U y 0 whi h implies that clos Zar ( S ) ∩ R k is on tained in the orthogonal of U y 0 . By denition of U y 0 , for all u ∈ U y 0 and all y ∈ S \ { y 0 } , the inner pro dut of − → y 0 u and − − → y 0 , y is zero. W e pro v e no w that the orthogonal of U y 0 is on tained in clos Zar ( S ) ∩ R k . By denition, the orthogonal of U y 0 is the set of lines L y ,y 0 for y ∈ S \ { y 0 } . Th us, it is suien t to pro v e that for all y ∈ S \ { y 0 } , L y ,y 0 is on tained in clos Zar ( S ) ∩ R k . F or all y ∈ S \ { y 0 } , S ∩ L y ,y 0 6 = ∅ b eause S is on v ex. Moreo v er, clos Zar ( S ∩ L y ,y 0 ) ∩ R k is L y ,y 0 . Sine S ∩ L y ,y 0 ⊂ S , L y ,y 0 is on tained in clos Zar ( S ) ∩ R k . Our assertion follo ws. Supp ose that dim( S ) = k . Then, b y Lemma 3.3 , S ∩ Q k is not empt y and a rational p oin t is outputted at Step (2) b y Corollary 2.6 . Supp ose no w that S is RR n ° 7045 12 Safey El Din & Zhi empt y . Then, an empt y list is returned at Step ( 4). W e supp ose no w that S is not empt y and that no rational p oin t is outputted at Step ( 6 ). Hene, w e en ter at Step (7). Remark that the form ula Θ (Step (7a )) denes the semi-algebrai set A ⊂ R k × R su h that ( a 1 , . . . , a k , b ) ∈ A if and only if ( a 1 , . . . , a k ) 6 = (0 , . . . , 0) and ∀ ( y 1 , . . . , y k ) ∈ R k ( y 1 , . . . , y k ) ∈ S = ⇒ a 1 y 1 + · · · + a k y k = b. Th us, the quan tier-free form ula W I i =1 Ψ i (Step (7b)) denes A . Note that b y Lemma 3.6 , A is not empt y . Hene, the lo op at Step ( 7) ends b y nding a rational parametrization G = ( G , G 0 , G 1 , . . . , G k , G k +1 ) (omputed at Step (7()i )) whi h eno des some p oin ts in A . From the sp eiation of SemiAlgeb raiSolve , G is irreduible o v er Q . Let a = ( a 1 , . . . , a k ) ∈ R k and b ∈ R su h that ( a, b ) ∈ A ∩ Z ( G ) . Then, there exists a real ro ot ϑ of G su h that G 0 ( ϑ ) a b = deg( G ) − 1 X i =1 ϑ i a i b i (4) where the ouples ( a i , b i ) ∈ Z k × Z are those returned b y Generate V eto rs (Step (7d)). Sine gcd( G 0 , G ) = 1 , G 0 ( ϑ ) 6 = 0 . Moreo v er, ( a, b ) ∈ A implies a 6 = (0 , . . . , 0) . Note also that ( a, b ) ∈ A implies that for all λ ∈ R ⋆ , ( λa, λb ) ∈ A sine for all ( y 1 , . . . , y k ) ∈ S and λ ∈ R ⋆ a 1 y 1 + · · · + a k y k = b ⇐ ⇒ λ ( a 1 y 1 + · · · + a k y k ) = λb This pro v es that ( a ⋆ , b ⋆ ) = ( G 0 ( ϑ ) a, G 0 ( ϑ ) b ) ∈ A and ( a ⋆ 1 , . . . , a ⋆ k ) 6 = (0 , . . . , 0) . Th us, there exists i su h that a i 6 = 0 , whi h implies that Step ( 7e) nev er fails. T o end the pro of of orretness, w e distinguish the ase where S ∩ Q k is empt y or not. The non-empt y ase. W e supp ose rst that S ∩ Q k is non-empt y; let ( y 1 , . . . , y k ) ∈ S ∩ Q k . Using (4), the linear relation a ⋆ 1 y 1 + · · · + a ⋆ k y k = b ⋆ implies the algebrai relation of degree deg( G ) − 1 : deg( G ) − 1 X i =0 ϑ i ( k X j =1 a i,j y j − b i ) = 0 , (5) where a i,j is the j -th o ordinate of a i . Sine G is irreduible, it is the minimal p olynomial of ϑ ; hene ϑ is an algebrai n um b er of degree deg( G ) . Th us, ( 5) is equiv alen t to ∀ 0 ≤ i ≤ deg( G ) − 1 , k X j =1 a i,j y j = b i . W e previously pro v ed that there exists i su h that a i 6 = 0 . W e let a = ( a 1 , . . . , a k ) ∈ Z k \ (0 , . . . , 0) and b ∈ Z b e resp etiv ely the v etor with in te- ger o ordinates and the in teger hosen in C (Step (7e)). W e ha v e just pro v ed INRIA R ational p oints in onvex semi-algebr ai sets 13 that S ∩ Q k is on tained in the in tersetion of S and of the ane h yp erplane H dened b y a 1 Y 1 + · · · + a k Y k = b . Note also that S ∩ H is on v ex sine S is on v ex and H is an ane h yp erplane. Consider the pro jetion π r : ( y 1 , . . . , y k ) ∈ R k → ( y 1 , . . . , y r − 1 , y r +1 , . . . , y k ) ∈ R k − 1 for the in teger r omputed at Step (7f). It is lear that the form ula Φ ′ omputed at Step (7h) denes the semi-algebrai set π r ( S ∩ H ) ⊂ R k − 1 . Sine S ∩ H is on v ex, π r ( S ∩ H ) is on v ex. Th us, the all to FindRationalP oints (Step (7i)) with inputs Φ ′ and [ Y 1 , . . . , Y r − 1 , Y r +1 , . . . , Y k ] is v alid. From the indution assumption, it returns a rational p oin t in π r ( S ∩ H ) if and only if π r ( S ∩ H ) has a non-empt y in tersetion with Q k − 1 . Sine S ∩ Q k (whi h is supp osed to b e non-empt y) is on tained in S ∩ H , π r ( S ∩ H ) on tains rational p oin ts. Th us, the list L (Step ( 7i)) on tains a rational p oin t q k − 1 = ( q 1 , . . . , q r − 1 , q r +1 , . . . , q k ) ∈ π r ( S ∩ H ) . This implies that π − 1 r ( q k − 1 ) ∩ H has a non-empt y in tersetion with S ∩ H . Remark that π − 1 r ( q k − 1 ) ∩ H is the rational p oin t q = ( q 1 , . . . , q r − 1 , q r , q r +1 , . . . , q k ) where q r is omputed at Step (7(j)ii ). It b elongs to S sine π − 1 r ( q k − 1 ) ∩ H and S ∩ H ha v e a non-empt y in tersetion. Th us, Φ( q 1 , . . . , q r − 1 , q r , q r +1 , . . . , q k ) is true and q is returned b y FindRationalP oints . The empt y ase. Supp ose no w that S ∩ Q k is empt y . As ab o v e H denotes the ane h yp erplane dened b y a 1 Y 1 + · · · + a k Y k = b where ( a 1 , . . . , a k ) ∈ Z k and b ∈ Z are hosen at Step (7e). Using the ab o v e argumen tation, π r ( S ∩ H ) is on v ex and the form ula Φ ′ (Step (7h)) denes π r ( S ∩ H ) . Th us, the all to FindRationalP oints (Step (7i )) with inputs Φ ′ and [ Y 1 , . . . , Y r − 1 , Y r +1 , . . . , Y k ] is v alid. Supp ose that π r ( S ∩ H ) do es not on tain rational p oin ts. Then, b y the indution assumption, L is empt y and the empt y list is returned (Step (7j)) whi h is the exp eted output sine w e ha v e supp osed S ∩ Q k = ∅ . Else, L on tains a rational p oin t ( q 1 , . . . , q r − 1 , q r +1 , . . . , q k ) . Consider the rational p oin t ( q 1 , . . . , q r − 1 , q r , q r +1 , . . . , q k ) (where q r is omputed at Step (7(j)ii )). It an not b elong to S sine w e ha v e supp osed S ∩ Q k is empt y . Consequen tly , Φ( q 1 , . . . , q r − 1 , q r , q r +1 , . . . , q k ) is false and the empt y list is returned. 4 Complexit y W e analyze no w the bit omplexit y of FindRationalP oints . Prop osition 4.1 Consider a set of p olynomials P = { h 1 , . . . , h s } ⊂ Z [ Y 1 , . . . , Y k ] , and a quantier-fr e e P -formula Φ( Y 1 , . . . , Y k ) and let D b e an inte ger suh that deg( h i ) ≤ D for 1 ≤ i ≤ s and σ the maximum bit length of the o eients of the h i 's. Then, FindRational Points (Φ , [ Y 1 , . . . , Y k ]) r e quir es σ O(1) ( sD ) O( k 3 ) bit op er ations. Mor e over, if it outputs a r ational p oint, its o or dinates have bit length dominate d by σ D O( k 3 ) . Remark 4.2 L et S ⊂ R k b e a onvex set dene d by S = { Y ∈ R k : R k ( Q 1 X [1] ∈ R n 1 ) · · · ( Q ω X [ ω ] ∈ R n ω ) P ( Y , X [1] , . . . , X [ ω ] ) } with quantiers Q i ∈ {∃ , ∀} , wher e X [ i ] is a set of n i variables, P is a Bo ole an funtion of s atomi pr e di ates g ( Y , X [1] , . . . , X [ ω ] ) ∆ i 0 RR n ° 7045 14 Safey El Din & Zhi wher e ∆ i ∈ { >, < , = } (for i = 1 , . . . , s ) and the g i 's ar e p olynomials of de gr e e D with inte ger o eients of binary size at most σ . Denote by Θ the quan- tie d formula dening S . By The or em 2.7 , QuantifierElimi nation (Θ) r e quir es σ s ( k +1)Π ω i =1 ( n i +1) D ( k +1)Π ω i =1 O( n i ) bit op er ations. It outputs a list of onjuntions Φ 1 , . . . , Φ I with I ≤ s ( k +1)Π ω i =1 ( n i +1) D ( k +1)Π ω i =1 O( n i ) , and for 1 ≤ i ≤ I , Φ i is a onjuntion of J i ≤ s Π ω i =1 ( n i +1) D Π ω i =1 O( n i ) atomi pr e di ates h ∆ 0 with h ∈ Z [ Y 1 , . . . , Y k ] , ∆ ∈ { = , > } and deg( h ) ≤ D Π ω i =1 O( n i ) and the bit length of the o eients of the p olynomials h ij is dominate d by σ D ( k +1)Π ω i =1 O( n i ) . Thus, the ost of running FindRationalP oints on al l the Φ i 's r e quir es σ O(1) ( sD ) O( k 3 Π ω i =1 n i ) bit op er ations. In ase of non-emptiness of S ∩ Q k , it r eturns an element of S ∩ Q k whose o or dinates have bit length dominate d by σ D O( k 3 Π ω i =1 n i ) . This ends to pr ove Cor ol lary 1.2 . W e start with a lemma. Lemma 4.3 Steps (1 -6) of FindRational Points (Φ) p erform within σ O(1) s k +1 D O( k ) bit op er ations. If a r ational p oint is r eturne d at Step (6 ) or Step (2), its o or di- nates have bit length dominate d by σ D O( k ) . Pr o of. The result is a diret onsequene of the results stated at Setion 2. 1. From Corollary 2.6 , Step (1) is p erformed within σ s k +1 D O( k ) bit op era- tions and if a rational p oin t is outputted at Step (2), its o ordinates ha v e bit length dominated b y σ D O( k ) . 2. From Prop osition 2.1 , Steps (3) and (4) are p erformed within σ O(1) s k +1 D O( k ) bit op erations. 3. From Prop osition 2.3 , Step (5) requires σ O(1) D O( k ) bit op erations. More- o v er, if a rational p oin t is outputted at Step (6), its o ordinates ha v e bit length dominated b y σ D O( k ) . W e pro v e no w the follo wing result. Lemma 4.4 1. Steps (7a -7h ) r e quir e σ O(1) ( sD ) O( k 2 ) bit op er ations. The numb er of p olynomials in Φ ′ is s ; their de gr e es ar e dominate d by D and the bit length of their o eients is dominate d by σ D O( k 2 ) . 2. If a r ational p oint with o or dinates of bit length dominate d by ℓ is r eturne d at Steps (7i -7j), the r ational numb er ompute d at Step (7(j)ii ) has bit length dominate d by ℓ + σ D O( k 2 ) . Pr o of. From Theorem 2.7 , Steps ( 7a -7b ) are p erformed within σ s O( k 2 ) D O( k 2 ) bit op erations. The obtained quan tier-free form ula is a disjuntion of ( sD ) O( k 2 ) onjuntions. Th us the lo op (Step ( 7)) mak es at most ( sD ) O( k 2 ) alls to SemiAl- geb raiSolve . Ea h onjuntions in v olv es ( sD ) O( k ) p olynomials of degree D O( k ) in Z [ A 1 , . . . , A k , B ] with in tegers of bit length dominated b y σ D O( k 2 ) . Th us, from Prop osition 2.1 , Step (7()i) is p erformed within σ O(1) ( sD ) O( k 2 ) bit op erations and outputs a rational parametrization of degree D O( k 2 ) with in teger o eien ts of bit length dominated b y σ D O( k 2 ) . Th us, the in tegers in the list omputed at Step ( 7d ) ha v e bit length dominated b y σ D O( k 2 ) . This INRIA R ational p oints in onvex semi-algebr ai sets 15 implies that the p olynomial obtained from Steps (7e-7g ) has rational o eien ts of bit length dominated b y σ D O( k 2 ) . Assertion (2) follo ws immediately . The bit omplexit y of these steps is ob viously negligible ompared to the ost of Step (7()i). The substitution phase (Step 7h) has a ost whi h is still dominated b y the ost of Step (7()i). As announed, the obtained form ula Φ ′ on tains s ( k − 1) -v ariate p olynomials of degree D with in teger o eien ts of bit length dominated b y σ D O( k 2 ) . W e pro v e no w Prop osition 4.1 b y indution on k . The initialization of the indution is immediate from Lemmata 3.5 and 4.3 . Supp ose that k > 1 . Supp ose that the exeution of FindRational Points (Φ) stops at Steps (2 ), or (4) or (6 ). From Lemma 4.3 , w e are done. Supp ose no w that w e en ter in Step (7). By Lemma 4.4 (1), the form ula Φ ′ omputed at Step (7h) on tains s ( k − 1) - v ariate p olynomials of degree D and o eien ts of bit length dominated b y σ D O( k 2 ) and is obtained within σ O(1) ( sD ) O( k 2 ) bit op erations. The indution assumption implies that Step (7i ) requires σ O(1) ( sD ) O( k 3 ) bit op erations, If a rational p oin t is on tained in L (Step ( 7j)), its o ordinates ha v e bit length dominated b y σ D O( k 3 ) . Hene, b y Lemma 4.4 (2), the rational n um b er omputed at Step ( 7(j)ii ) has bit length dominated b y σ D O( k 3 ) . Moreo v er, the ost of Steps (7(j)ii -7(j)iii ) is negligible ompared to the ost of previous steps. 5 Rational sums of squares Consider a p olynomial f ∈ Z [ x 1 , . . . , x n ] of degree 2 d whose o eien ts ha v e bit length b ounded b y τ . If w e ho ose v as the v etor of all monomials in Z [ x 1 , . . . , x n ] of degree less than or equal to d , then w e onsider the set of real symmetri matries M = M T of dimension D = n + d n for whi h f = v T · M · v . By Gaussian elimination, it follo ws that there exists an in teger k ≤ 1 2 D ( D + 1) − n +2 d n su h that M = { M 0 + Y 1 M 1 + . . . + Y k M k , Y 1 , . . . , Y k ∈ R } (6) for some rational symmetri matries M 0 , . . . , M k . The p olynomial f an b e written as a sum of squares of p olynomials if and only if the matrix M an b e ompleted as a symmetri p ositiv e semiden te matrix (see Lauren t (2001 )). Let Y = ( Y 1 , . . . , Y k ) , w e dene S = { Y ∈ R k | M ( Y ) 0 , M ( Y ) = M ( Y ) T , f = v T · M ( Y ) · v } . (7) It is lear that S ⊆ R k is a on v ex set dened b y setting all p olynomials in Φ( Y 1 , . . . , Y k ) = { ( − 1 ) ( i + D ) m i , i = 0 , . . . , D − 1 } (8) to b e nonnegativ e, where the m i 's are the o eien ts of the harateristi p oly- nomial of M ( Y ) . The ardinalit y s of Φ is b ounded b y D and Φ on tains p oly- nomials of degree b ounded b y D whose o eien ts ha v e bit length b ounded b y RR n ° 7045 16 Safey El Din & Zhi τ D (see P o w ers and Wörmann (1998 )). Hene the semi-algebrai set dened b y (7) is S = { ( Y 1 , . . . , Y k ) ∈ R k | ( − 1) ( i + D ) m i ≥ 0 , 0 ≤ i ≤ D − 1 } . (9) The result b elo w is obtained b y applying Theorem 1.1 to the semi-algebrai set dened ab o v e. Corollary 5.1 L et f ∈ Z [ x 1 , . . . , x n ] of de gr e e 2 d with inte gers of bit length b ounde d by τ . By running the algorithm FindRationalP oints for the semi-algebr ai set dene d in (7), one an de ide whether f is a sum of squar es in Q [ x 1 , . . . , x n ] within τ O(1) D O( k 3 ) bit op er ations. Supp ose f = P f 2 i , f i ∈ Q [ x 1 , . . . , x n ] , then the bit lengths of r ational o eients of the f i 's ar e b ounde d by τ D O( k 3 ) . Remark 5.2 Applying Pr op osition 1.3 by Khahiyan and Porkolab to the semi- algebr ai set dene d in (7), one an de ide whether f is a sum of squar es in Z [ x 1 , . . . , x n ] within τ O(1) D O( k 4 ) op er ations. Supp ose that f = P f 2 i , f i ∈ Z [ x 1 , . . . , x n ] , then the bit lengths of inte ger o eients of f i ar e b ounde d by τ D O( k 4 ) . P ork olab and Kha hiy an sho w ed that the non-emptiness of the on v ex set dened in (7) o v er the reals an b e determined in O( k D 4 ) + D O(min { k,D 2 } ) arithmeti op erations o v er ℓD O(min { k,D 2 } ) -bit n um b ers, where ℓ is the maximal bit length of the matries M i (see P ork olab and Kha hiy an (1997 )). Supp ose S 6 = ∅ , i.e., f ∈ Q [ x 1 , . . . , x n ] is a sum of m squares in K [ x 1 , . . . , x n ] where K is an algebrai extension of Q . If K is a totally real n um b er eld, then f is also a sum of squares in Q [ x 1 , . . . , x n ] , i.e, S T Q n 6 = ∅ (see Hillar (2009 ); Kaltofen (2009 )). The follo wing lemma and pro of an b e dedued from argumen ts giv en in Kaltofen (2009 ). Lemma 5.3 Supp ose G = ( G, G 0 , G 1 , . . . , G k ) is a r ational p ar ametrization for the semi-algebr ai set S dene d in (7) ompute d by SemiAlgeb raiSolve . Supp ose ϑ is a r e al r o ot of G suh that Y ( ϑ ) = 1 q ( G 1 ( ϑ ) , G 2 ( ϑ ) , . . . , G k ( ϑ )) ∈ S , (10) Then for any r e al r o ot ϑ i of G , we have Y ( ϑ i ) = 1 q ( G 1 ( ϑ i ) , G 2 ( ϑ i ) , . . . , G k ( ϑ i )) ∈ S . (11) Mor e over, if the p olynomial G has only r e al r o ots, then the p oint dene d by 1 deg G P deg G i =1 Y ( ϑ i ) is a r ational p oint in S . Pr o of. Sine Y ( ϑ ) ∈ S , the matrix M ( Y ( ϑ )) is p ositiv e semidenite. W e an p erform the Gaussian elimination o v er Q ( ϑ ) to obtain the deomp osition M ( Y ( ϑ )) = A ( ϑ ) T A ( ϑ ) . It is lear that for an y real ro ot ϑ i of G , M ( Y ( ϑ i )) = A ( ϑ i ) T A ( ϑ i ) is also p ositiv e semi-denite, i.e., Y ( ϑ i ) ∈ S . Moreo v er, if G has only real ro ots ϑ i , then P ϑ i ,G ( ϑ i )=0 G j ( ϑ i ) ∈ Q . It follo ws that the p oin t dened b y 1 deg G P deg G i =1 Y ( ϑ i ) is a rational p oin t in S . The ab o v e disussion leads to the follo wing result. INRIA R ational p oints in onvex semi-algebr ai sets 17 Theorem 5.4 Supp ose f ∈ Z [ x 1 , . . . , x n ] . Ther e exists a funtion RationalT o- talRealSolve whih either determines that f an not b e written as sum of squar es over the r e als or r eturns a sum of squar es r epr esentation of f over Q [ x 1 , . . . , x n ] if and only if the p olynomial G outputte d fr om the funtion SemiAlgeb raiSolve has only r e al solutions. The o or dinates of the r ational o eients of p olynomi- als f i in f = P i f 2 i have bit length dominate d by τ D O( k ) and the bit omplexity of RationalT otalRealSolve is τ O(1) D O( k ) . Referenes Basu, S., P olla k, R., Ro y , M.-F., 1996. On the om binatorial and algebrai omplexit y of quan tier elimination. J. A CM 43 (6), 10021045. Basu, S., P olla k, R., Ro y , M.-F., 2006. Algorithms in real algebrai geome- try , 2nd Edition. V ol. 10 of Algorithms and Computation in Mathematis. Springer-V erlag. Bo hnak, J., Coste, M., Ro y , M.-F., 1998. Real Algebrai Geometry . Springer- V erlag. Choi, M., Lam, T., Rezni k, B., 1995. Sums of squares of real p olynomials. Symp. in Pure Math. 58 (2), 103126. Heinz, S., 2005. Complexit y of in teger quasion v ex p olynomial optimization. J. Complex. 21 (4), 543556. Hillar, C., 2009. Sums of p olynomial squares o v er totally real elds are rational sums of squares. Pro . Amerian Math. So iet y 137, 921930. Kaltofen, E., Li, B., Y ang, Z., Zhi, L., 2008. Exat ertiation of global op- timalit y of appro ximate fatorizations via rationalizing sums-of-squares with oating p oin t salars. In: Pro . ISSA C08. pp. 155163. Kaltofen, E., Li, B., Y ang, Z., Zhi, L., jan 2009. Exat ertiation in global p olynomial optimization via sums-of-squares of rational funtions with ratio- nal o eien ts. Man usript, 20 pages. Kaltofen, E. L., 2009. Priv ate omm uniation, F ebruary 24, 2009. Kha hiy an, L., P ork olab, L., 1997. Computing in tegral p oin ts in on v ex semi- algebrai sets. F oundations of Computer Siene, Ann ual IEEE Symp osium on 0, 162171. Kha hiy an, L., P ork olab, L., 2000. In teger optimization on on v ex semialgebrai sets. Disrete and Computational Geometry 23 (2), 207224. Lauren t, M., 2001. P olynomial instanes of the p ositiv e semidenite and eu- lidean distane matrix ompletion problems. SIAM J. Matrix Anal. Appl. 22 (3), 874894. Lenstra, A. K., H. W. Lenstra, H. W., Lo vàsz, L., 1982. F atoring p olynomials with rational o eien ts. Math. Ann. 261, 515534. RR n ° 7045 18 Safey El Din & Zhi Lenstra, H. W., J., 1983. In teger programming with a xed n um b er of v ariables. Mathematis of Op erations Resear h 8 (4), 538548. URL http://www.jstor .or g/ st abl e/ 36 891 68 Mignotte, M., 1982. Some useful b ounds. In: Bu h b erger, B., Collins, G. E., Lo os, R. (Eds.), Computer Algebra, Sym b oli and Algebrai Computation. Supplemen tum to Computing. Springer V erlag, pp. 259263. P eyrl, H., P arrilo, P . A., 2007. A Maaula y 2 pa k age for omputing sum of squares deomp ositions of p olynomials with rational o eien ts. In: Pro . SNC'07. pp. 207208. P eyrl, H., P arrilo, P . A., 2008. Computing sum of squares deomp ositions with rational o eien ts. Theoretial Computer Siene 409, 269281. P ork olab, L., Kha hiy an, L., 1997. On the omplexit y of semidenite programs. J. of Global Optimization 10 (4), 351365. P o w ers, V., Wörmann, T., 1998. An algorithm for sums of squares of real p oly- nomials. Journal of Pure and Applied Algebra 6 (1), 99104. S hönhage, A., 1984. F atorization of univ ariate in teger p olynomials b y diophan- tine apro ximation and an impro v ed basis redution algorithm. In: ICALP . pp. 436447. v an Ho eij, M., No v o in, A., 2007. Complexit y results for fator- ing univ ariate p olynomials o v er the rationals. Preprin t, URL: http://www.math.f su .e du/ hoeij/papers/20 07/ pa pe r6. pd f . 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