7. The Algebraic-Geometric Dictionary Equality constraints Ideals - PowerPoint PPT Presentation

7 - 1 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 7. The Algebraic-Geometric Dictionary • Equality constraints • Ideals and Varieties • Feasibility problems and duality • The Nullstellensatz and strong duality • The B´ ezout identity and fundamental theorem of algebra • Partition of unity • Certificates • Abstract duality • The ideal-variety correspondence • Computation and Groebner bases • Real variables and inequalities

7 - 2 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Equality Constraints Consider the feasibility problem does there exist x ∈ R n such that f i ( x ) = 0 for all i = 1 , . . . , m The function f : R n → R is called a valid equality constraint if f ( x ) = 0 for all feasible x Given a set of equality constraints, we can generate others as follows. (i) If f 1 and f 2 are valid equalities, then so is f 1 + f 2 (ii) For any h ∈ R [ x 1 , . . . , x n ] , if f is a valid equality, then so is hf

7 - 3 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 The Ideal of Valid Equality Constraints A set of polynomials I ⊂ R [ x 1 , . . . , x n ] is called an ideal if (i) f 1 + f 2 ∈ I for all f 1 , f 2 ∈ I (ii) fh ∈ I for all f ∈ I and h ∈ R [ x 1 , . . . , x n ] • Given f 1 , . . . , f m , we can generate an ideal of valid equalities by re- peatedly applying these rules. • This gives the ideal generated by f 1 , . . . , f m , written ideal { f 1 , . . . , f m } . � m � � ideal { f 1 , . . . , f m } = h i f i | h i ∈ R [ x 1 , . . . , x n ] i =1 This is also written � f 1 , . . . , f m � . • Every polynomial in ideal { f 1 , . . . , f m } is a valid equality.

7 - 4 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 More on Ideals • For S ⊂ R n , the ideal of S is � � I ( S ) = f ∈ R [ x 1 , . . . , x n ] | f ( x ) = 0 for all x ∈ S • ideal { f 1 , . . . , f m } is the smallest ideal containing f 1 , . . . , f m . The polynomials f 1 , . . . , f m are called the generators of the ideal. • If I 1 and I 2 are ideals, then so is I 1 ∩ I 2 • Every ideal in R [ x 1 , . . . , x n ] is finitely generated. (This does not hold for non-commutative polynomials) • An ideal generated by one polynomial is called a principal ideal.

7 - 5 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Varieties We’ll need to work over both R and C ; we’ll use K to denote either. The variety defined by polynomials f 1 , . . . , f m ∈ K [ x 1 , . . . , x m ] is x ∈ K n | f i ( x ) = 0 for all i = 1 , . . . , m � � V{ f 1 , . . . , f m } = A variety is also called an algebraic set . • V{ f 1 , . . . , f m } is the set of all solutions x to the feasibility problem f i ( x ) = 0 for all i = 1 , . . . , m

7 - 6 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Examples of Varieties • If f ( x ) = x 2 1 + x 2 2 − 1 then V ( f ) is the unit circle in R 2 . • The graph of a polynomial function h : R → R is the variety of f ( x ) = x 2 − h ( x 1 ) . • The affine set x ∈ R n | Ax = b � � is the variety of the polynomials a T i x − b i

7 - 7 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Properties of Varieties • If V, W are varieties, then so is V ∩ W because if V = V{ f 1 , . . . , f m } and W = V{ g 1 , . . . , g n } then V ∩ W = V{ f 1 , . . . , f m , g 1 , . . . , g n } • so is V ∪ W , because � � V ∪ W = V f i g j | i = 1 , . . . , m, j = 1 , . . . , n • If V is a variety, the projection of V onto a subspace may not be a variety. • The set-theoretic difference of two varieties may not be a variety.

7 - 8 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Feasibility Problems and Duality Suppose f 1 , . . . , f m are polynomials, and consider the feasibility problem does there exist x ∈ K n such that f i ( x ) = 0 for all i = 1 , . . . , m Every polynomial in ideal { f 1 , . . . , f m } is zero on the feasible set. So if 1 ∈ ideal { f 1 , . . . , f m } , then the primal problem is infeasible. Again, this is proof by contradiction. Equivalently, the primal is infeasible if there exist polynomials h 1 , . . . , h m ∈ K [ x 1 , . . . , x n ] such that for all x ∈ K n 1 = h 1 ( x ) f 1 ( x ) + · · · + h m ( x ) f m ( x )

7 - 9 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Strong Duality So far, we have seen examples of weak duality. The Hilbert Nullstellensatz gives a strong duality result for polynomials over the complex field. The Nullstellensatz Suppose f 1 , . . . , f m ∈ C [ x 1 , . . . , x n ] . Then 1 ∈ ideal { f 1 , . . . , f m } ⇐ ⇒ V C { f 1 , . . . , f m } = ∅

7 - 10 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Algebraically Closed Fields For complex polynomials f 1 , . . . , f m ∈ C [ x 1 , . . . , x n ] , we have 1 ∈ ideal { f 1 , . . . , f m } ⇐ ⇒ V{ f 1 , . . . , f m } = ∅ This does not hold for polynomials and varieties over the real numbers. For example, suppose f ( x ) = x 2 + 1 . Then � � V R { f } = x ∈ R | f ( x ) = 0 = ∅ But 1 �∈ ideal { f } , since any multiple of f will have degree ≥ 2 . The above results requires an algebraically closed field . Later, we will see a version of this result that holds for real varieties.

7 - 11 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 The Nullstellensatz and Feasibility Problems The primal problem: does there exist x ∈ C n such that f i ( x ) = 0 for all i = 1 , . . . , m The dual problem: do there exist h 1 , . . . , h m ∈ C [ x 1 , . . . , x n ] such that 1 = h 1 f 1 + · · · + h m f m The Nullstellensatz implies that these are strong alternatives . Exactly one of the above problems is feasible.

7 - 12 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Example: Nullstellensatz Consider the polynomials f 1 ( x ) = x 2 f 2 ( x ) = 1 − x 1 x 2 1 There is no x ∈ C 2 which simultaneously satisfies f 1 ( x ) = 0 and f 2 ( x ) = 0 ; i.e., V{ f 1 , f 2 } = ∅ Hence the Nullstellensatz implies there exists h 1 , h 2 such that 1 = h 1 ( x ) f 1 ( x ) + h 2 ( x ) f 2 ( x ) One such pair is h 1 ( x ) = x 2 h 2 ( x ) = 1 + x 1 x 2 2

7 - 13 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Interpretations of the Nullstellensatz • The feasibility question asks; do the polynomials f 1 , . . . , f m have a common root ? The Nullstellensatz is a B´ ezout identity . In the scalar case, the dual problem is: do the polynomials have a common factor ? • Suppose we look at f ∈ C [ x ] , a scalar polynomial with complex coef- ficients. The feasibility problem is: does it have a root? The Nullstellensatz says it has a root if and only if there is no polyno- mial h ∈ C [ x ] such that 1 = hf Since degree ( hf ) ≥ degree ( f ) , there is no such h if degree ( f ) ≥ 1 ; i.e. all polynomials f with degree ( f ) ≥ 1 have a root. So the Nullstellensatz generalizes the fundamental theorem of algebra.

7 - 14 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Interpretation: Partition of Unity The equation 1 = h 1 f 1 + · · · + h m f m is called a partition of unity . For example, when m = 2 , we have 1 = h 1 ( x ) f 1 ( x ) + h 2 ( x ) f 2 ( x ) for all x � � x ∈ C n | f i ( x ) = 0 Let V i = . Let q ( x ) = h 1 ( x ) f 1 ( x ) . Then for x ∈ V 1 , we have q ( x ) = 0 , and hence the second term h 2 ( x ) f 2 ( x ) equals one. Conversely, for x ∈ V 2 , we must have q ( x ) = 1 . Since q ( x ) cannot be both zero and one, we must have V 1 ∩ V 2 = ∅ .

7 - 15 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Interpretation: Certificates The functions h 1 , . . . , h m give a certificate of infeasibility for the primal problem. Given the h i , one may immediately computationally verify that 1 = h 1 f 1 + · · · + h m f m and this proves that V{ f 1 , . . . , f m } = ∅

7 - 16 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 Duality The notion of duality here is parallel to that for linear functionals. Compare, for S ⊂ R n � � � f ( x ) = 0 for all x ∈ S � I ( S ) = f ∈ R [ x 1 , . . . , x n ] with S ⊥ = � � p ∈ ( R n ) ∗ � � � p, x � = 0 for all x ∈ S • There is a pairing between R n and ( R n ) ∗ ; we can view either as a space of functionals on the other • The same holds between R n and R [ x 1 , . . . , x n ] • If S ⊂ T , then S ⊥ ⊃ T ⊥ and I ( S ) ⊃ I ( T )

7 - 17 The Algebraic-Geometric Dictionary P. Parrilo and S. Lall 2006.06.07.01 The Ideal-Variety Correspondence Given S ⊂ K n , we can construct the ideal � � � f ( x ) = 0 for all x ∈ S � I ( S ) = f ∈ K [ x 1 , . . . , x n ] Also given an ideal I ⊂ K [ x 1 , . . . , x n ] we can construct the variety x ∈ K n | f ( x ) = 0 for all f ∈ I � � V ( I ) = If S is a variety, then � � V I ( S ) = S This implies I is one-to-one (since V is a left-inverse); i.e., no two distinct varieties give the same ideal.