Computing Rational Points in Convex Semi-algebraic Sets and Sums-of-Squares Decompositions

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1 Computing Rational Points in Convex Semi-algebraic Sets and Sums-of-Squares Decompositions Mohab Safey El Din 1 Lihong Zhi 2 1 University Pierre et Marie Curie, Paris 6, France INRIA Paris-Rocquencourt, SALSA Project-Team Laboratoire d Informatique de Paris 6 2 Mathematics Mechanization Research Center, Academy of Mathematics and System Sciences, China Workshop GeoLMI Rennes, 2011

2 Problem statement Let S R n be a semi-algebraic set defined by f 1 = = f p = 0, g 1 σ 1 0,..., g k σ k 0 with f i and g i in Q[X 1,..., X n ] and σ i {>, }. In the sequel we suppose that S is convex.

3 Problem statement Let S R n be a semi-algebraic set defined by f 1 = = f p = 0, g 1 σ 1 0,..., g k σ k 0 with f i and g i in Q[X 1,..., X n ] and σ i {>, }. In the sequel we suppose that S is convex. Problem 1: Decidability Emptiness of S Q n (and compute points) Algorithm?

4 Problem statement Let S R n be a semi-algebraic set defined by f 1 = = f p = 0, g 1 σ 1 0,..., g k σ k 0 with f i and g i in Q[X 1,..., X n ] and σ i {>, }. In the sequel we suppose that S is convex. Problem 1: Decidability Emptiness of S Q n (and compute points) Algorithm? Problem 2: Complexity What is its bit complexity? Bounds on the height of the outputted points (w.r.t degree, height of coefficients, of the input)?

5 Example Let S R 3 be the semi-algebraic set defined by S is convex. X 1 = X 2 X 3, X = 0, X 3 > 0, X 1 > 0 S Q 3 = because {(x 1, x 2 ) x 1 = 2x 2 } Q 2 = {0}

6 Motivations Sums-of-Squares Decompositions and LMI s: 2D = deg(f), v = Monomials(D) and M a symetric linear matrix s.t. f = v t Mv. (g 1,..., g s ) f = g g 2 s y f = v t M y v with M y 0 Linear Matrix Inequality (LMI): M 0 + Y 1.M 1 + Y N.M N 0 This LMI defines a convex semi-algebraic set S R N.

7 Motivations Sums-of-Squares Decompositions and LMI s: 2D = deg(f), v = Monomials(D) and M a symetric linear matrix s.t. f = v t Mv. (g 1,..., g s ) f = g g 2 s y f = v t M y v with M y 0 Linear Matrix Inequality (LMI): M 0 + Y 1.M 1 + Y N.M N 0 This LMI defines a convex semi-algebraic set S R N. Sturmfels conjecture Let f Q[X 1,..., X n ] s.t. f = g g2 s (with g i R[X 1,..., X n ]). Then there exists h 1,..., h p in Q[X 1,..., X n ] s.t. f = h h2 p. f = p i=1 h2 i with h i Q[X 1,..., X n ] S Q N

8 State of the art (I) Existence of sums-of-squares decompositions over the rationals. C. Hillar (Proc. of the AMS, 2009) Let K be a totally real number field. Suppose f = i g2 i with coefficients of g i in K. Then, f can be decomposed into SOS with coefficients in Q. Other (simpler) proofs due to Kaltofen and Quarez.

9 State of the art (II) Kaltofen, Li, Yang, Zhi (series of papers in ISSAC, JSC) (see also Parillo, Peyrl SNC) Symbolic/numeric approach: numerical LMI-solvers rational reconstruction of the coefficients Applications: Rump s model problems Monotone permanent conjecture Surprising facts Growth of the coefficients in the SOS decompositions Are we sure that one can always reconstruct rational solutions to the LMI? rational points in convex semi-algebraic sets? bounds on the bit-size?

10 State of the art (III) Convex semi-algebraic sets and integer programming. Kachiyan/Porkolab (FOCS DCG 2000) Algorithm for deciding if S Z n, compute integral points in S, complexity and bounds on the height of the outputted points. all what we want to do over the rationals has been done over the integers!

11 State of the art (III) Convex semi-algebraic sets and integer programming. Kachiyan/Porkolab (FOCS DCG 2000) Algorithm for deciding if S Z n, compute integral points in S, complexity and bounds on the height of the outputted points. all what we want to do over the rationals has been done over the integers! Key tools: Quantifier elimination over the reals, Sampling points in semi-algebraic sets, Basic elements of number theory and algebra.

12 Main Results P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N convex defined by a P-formula Φ(Y 1,..., Y N ).

13 Main Results P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N convex defined by a P-formula Φ(Y 1,..., Y N ). First result: Algorithm and Complexity It decides if S Q N within σ O(1) (sδ) O(N 3) bit operations. If S Q N, it returns a point in S Q N whose coordinates have bit length σδ O(N 3).

14 Main Results P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N convex defined by a P-formula Φ(Y 1,..., Y N ). First result: Algorithm and Complexity It decides if S Q N within σ O(1) (sδ) O(N 3) bit operations. If S Q N, it returns a point in S Q N whose coordinates have bit length σδ O(N 3). Second result: Quantitative result for SOS decompositions over Q Let f Z[X 1,..., X n ] with coefficients of bit size τ and deg(f) = 2D. One can decide if f = fi 2, f i Q[X 1,..., X n ] within τ O(1) D O(n3) bit operations. The bit lengths of rational coefficients of the f i s are bounded by τδ O(N 3 ) with δ = ( n+d n ) and N 1 2 δ(δ + 1) ( n+2d n )

15 Consequences of convexity P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N defined by a P-formula and convex. Either dim(s ) = N, or there exists a hyperplane H R N such that S H. In the sequel, H S denotes the set of such hyperplanes. Our Example S defined by: X 1 = X 2 X 3, X = 0, X 3 > 0, X 1 > 0 H S is the set of hyperplanes containing the vector ( 2, 1, 2).

16 Basic tools P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N defined by a P-formula.

17 Basic tools P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N defined by a P-formula. Sampling points in semi-algebraic sets Best complexity results due to Basu/Pollack/Roy σ O(1) s N δ O(N) bit operations size of the output σδ O(N) Extensions to compute sampling points in the interior of S.

18 Basic tools P = {f 1,..., f s } Z[Y 1,..., Y N ] with coefficients of bit length σ and degrees δ. S R N defined by a P-formula. Sampling points in semi-algebraic sets Best complexity results due to Basu/Pollack/Roy σ O(1) s N δ O(N) bit operations size of the output σδ O(N) Extensions to compute sampling points in the interior of S. Quantifier elimination Best complexity results due to Basu/Pollack/Roy: Bit-complexity and size of the output doubly exponential in the number of alternates of quantifiers

19 Algorithm (I) FindRationalPoints. Input: Φ(Y 1,..., Y N ) defining a convex semi-algebraic set S Output: a list of rational points in S Q n iff it is not empty. The algorithm is recursive in N

20 Algorithm (I) FindRationalPoints. Input: Φ(Y 1,..., Y N ) defining a convex semi-algebraic set S Output: a list of rational points in S Q n iff it is not empty. The algorithm is recursive in N First steps. Compute sample points in S (useful if dim(s ) 0). done within σ O(1) (sδ) O(N) bit operations output of size σδ O(N)

21 Algorithm (I) FindRationalPoints. Input: Φ(Y 1,..., Y N ) defining a convex semi-algebraic set S Output: a list of rational points in S Q n iff it is not empty. The algorithm is recursive in N First steps. Compute sample points in S (useful if dim(s ) 0). done within σ O(1) (sδ) O(N) bit operations output of size σδ O(N) Compute sample points in the interior of S (useful if dim(s ) = N). done within σ O(1) (sδ) O(N) bit operations output of size σδ O(N)

22 Algorithm (II) (a) Solve the QE problem defining the semi-algebraic set H S y R N y S = N a i y i = b and i=1 N a 2 i 0 i=1 One gets a formula Ψ defining H S done within σ(sδ) O(N 2) bit operations disjunction of (sδ) O(N 2) conjunctions involving (sδ) O(N) polys of degree δ O(N) coeffs of size σδ O(N)

23 Algorithm (II) (a) Solve the QE problem defining the semi-algebraic set H S y R N y S = N a i y i = b and i=1 N a 2 i 0 i=1 One gets a formula Ψ defining H S done within σ(sδ) O(N 2) bit operations disjunction of (sδ) O(N 2) conjunctions involving (sδ) O(N) polys of degree δ O(N) coeffs of size σδ O(N) (b) Compute sample points in H S encoded by rational parametrizations q(ϑ) = 0, a i = q i (ϑ)/q 0 (ϑ) for 1 i N, b = q N+1 (ϑ)/q 0 (ϑ) with q irreducible and deg(q i ) deg(q) 1 for 0 i N + 1 done within σ O(1) (sd) O(N 2) bit operations coeffs of size σd O(N 2 )

24 Algorithm (III) (c) Let (y 1,..., y N ) S Q N, q 1 (ϑ)y q N (ϑ)y N = q N+1 (ϑ) can be rewritten as deg(q) 1 i=0 N ϑ i ( q i,j y j q i,n+1 ) = 0 j=1 N j=1 q i,jy j q i,n+1 = 0 is satisfied by all points in S Q N Use one non-trivial such linear equation and to eliminate one coordinate in Φ one gets Φ which still defines a convex semi-algebraic set

25 Algorithm (III) (c) Let (y 1,..., y N ) S Q N, q 1 (ϑ)y q N (ϑ)y N = q N+1 (ϑ) can be rewritten as deg(q) 1 i=0 N ϑ i ( q i,j y j q i,n+1 ) = 0 j=1 N j=1 q i,jy j q i,n+1 = 0 is satisfied by all points in S Q N Use one non-trivial such linear equation and to eliminate one coordinate in Φ one gets Φ which still defines a convex semi-algebraic set (d) Recursive call to FindRationalPoints with input Φ the number of variables has decreased by 1 Degrees are the same in Φ as in Φ, bit length of coefficients has been multiplied by σδ O(N 2 )

26 Back to sums-of-squares f Z[X 1,..., X n ] s.t. deg(f) = 2D and bit length of coefficients τ. v is the vector of all monomials in Z[X 1,..., X n ] of degree D,

27 Back to sums-of-squares f Z[X 1,..., X n ] s.t. deg(f) = 2D and bit length of coefficients τ. v is the vector of all monomials in Z[X 1,..., X n ] of degree D, N 1 2 δ(δ + 1) ( ) n+2d n s.t. E = {S(y) = S 0 N i=0 +y i S i, y R N v t S(y)v = f} for some rational symmetric matrices S 0,..., S N. f is a SOS iff y such that S(y) = S 0 + y 1 S y N S N 0.

28 Back to sums-of-squares f Z[X 1,..., X n ] s.t. deg(f) = 2D and bit length of coefficients τ. v is the vector of all monomials in Z[X 1,..., X n ] of degree D, N 1 2 δ(δ + 1) ( ) n+2d n s.t. E = {S(y) = S 0 N i=0 +y i S i, y R N v t S(y)v = f} for some rational symmetric matrices S 0,..., S N. f is a SOS iff y such that S(y) = S 0 + y 1 S y N S N 0. Let S = {y R n S(y) 0, S(y) = S(y) T, f = v T S(y) v}. S R N is a convex semi-algebraic set defined by Φ(Y 1,..., Y n ) = {( 1) (i+δ) m i 0, i = 0,..., δ 1} where the m i s are the coefficients of the characteristic polynomial of S(Y ). s δ and deg(m i ) δ whose coefficients have bit length τδ. Then, we apply the complexity analysis of FindRationalPoints to this input.

29 Conclusions/Perspectives First algorithm for computing rational points in convex semi-algebraic sets and first bounds on the bit-size of the outputted points based on Kachiyan/Porkolab s ideas.

30 Conclusions/Perspectives First algorithm for computing rational points in convex semi-algebraic sets and first bounds on the bit-size of the outputted points based on Kachiyan/Porkolab s ideas. Optimality of the bound/algorithm? probably no, the recursion should not be more than dim(s ) and may be also reduced on-going work: N 2 in the exponent instead of N 3. Open Problems: N in the exponent? Exploit convexity in symbolic/exact algorithms?

31 Conclusions/Perspectives First algorithm for computing rational points in convex semi-algebraic sets and first bounds on the bit-size of the outputted points based on Kachiyan/Porkolab s ideas. Optimality of the bound/algorithm? probably no, the recursion should not be more than dim(s ) and may be also reduced on-going work: N 2 in the exponent instead of N 3. Open Problems: N in the exponent? Exploit convexity in symbolic/exact algorithms? Specific bounds for LMI? LMI define semi-algebraic sets which are highly structured (defined by minors of linear matrices) see Faugère/S./Spaenlehauer ISSAC 2010 for specific properties of ideals generated by such systems

COMPUTING RATIONAL POINTS IN CONVEX SEMI-ALGEBRAIC SETS AND SOS DECOMPOSITIONS

COMPUTING RATIONAL POINTS IN CONVEX SEMI-ALGEBRAIC SETS AND SOS DECOMPOSITIONS MOHAB SAFEY EL DIN AND LIHONG ZHI Abstract. Let P = {h 1,..., h s} Z[Y 1,..., Y k ], D deg(h i ) for 1 i s, σ bounding the