Real Root Finding for Equivariant Semi-Algebraic Systems ISSAC 2018 - PowerPoint PPT Presentation

Real Root Finding for Equivariant Semi-Algebraic Systems ISSAC 2018 Cordian Riener 1 Mohab Safey El Din 2 1 UiT - The Arctic University of Norway 2 Sorbonne Universit´ e, CNRS , INRIA Laboratoire d’Informatique de Paris 6, LIP6 , ´ Equipe PolSys July 9, 2018 C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 1 / 17

Introduction The objects we are looking at and the questions one can ask In the sequel we will use a real closed field R C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 2 / 17

Introduction The objects we are looking at and the questions one can ask In the sequel we will use a real closed field R a finite set P ⊂ R [ X 1 , . . . , X k ] defining a real variety Z R ( P ) ⊂ R k , C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 2 / 17

Introduction The objects we are looking at and the questions one can ask In the sequel we will use a real closed field R a finite set P ⊂ R [ X 1 , . . . , X k ] defining a real variety Z R ( P ) ⊂ R k , or more generally a semi-algebraic S ⊂ R k , which can be described by P . Algorithmic problems associated to these, like, e.g., real root finding, connectivity queries, quantifier elimination, are known to be intrinsically hard. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 2 / 17

Symmetry Philosophy: structure in a set can reduce complexity C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 3 / 17

How to exploit Symmetry ? Do general symmetric problems have symmetric solutions? YES, this must be the case! NO, this is just so wrong! C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 4 / 17

How to exploit Symmetry ? Do general symmetric problems have symmetric solutions? YES, this must be the case! NO, this is just so wrong! Proposition Let S ⊂ R n be basic convex symmetric semi-algebraic set. Then S is not empty if and only if it contains a point for which all coordinates are equal. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 4 / 17

What to do in a non convex situation? Previous results Theorem Let { f , f 1 , . . . , f s } ⊂ R [ x 1 , . . . , x n ] be Sym n -invariant polynomials of degree at most d. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 5 / 17

What to do in a non convex situation? Previous results Theorem Let { f , f 1 , . . . , f s } ⊂ R [ x 1 , . . . , x n ] be Sym n -invariant polynomials of degree at most d. The real algebraic set V ( f ) is not empty if and only if it contains a point 1 with at most ⌊ d 2 ⌋ distinct coordinates. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 5 / 17

What to do in a non convex situation? Previous results Theorem Let { f , f 1 , . . . , f s } ⊂ R [ x 1 , . . . , x n ] be Sym n -invariant polynomials of degree at most d. The real algebraic set V ( f ) is not empty if and only if it contains a point 1 with at most ⌊ d 2 ⌋ distinct coordinates. The semi-algebraic set in S ⊂ R n defined by f 1 ≥ 0 , . . . , f s ≥ 0 is not empty if 2 and only if it contains a point with at most d distinct coordinates. This result helps to drastically reduce complexity and in particular it allows for a polynomial (in n ) algorithm to test, if a given system described by symmetric polynomials of fixed degree has a solution. But what can we say in more general situations? C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 5 / 17

A more general situation Definition Consider G = ( g 1 , . . . , g n ) a sequences of polynomials in R [ x 1 , . . . , x n ]. If for all σ ∈ S n , we have G ( σ ( X )) = σ ( G )( X ) , then G is called equivariant. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 6 / 17

A more general situation Definition Consider G = ( g 1 , . . . , g n ) a sequences of polynomials in R [ x 1 , . . . , x n ]. If for all σ ∈ S n , we have G ( σ ( X )) = σ ( G )( X ) , then G is called equivariant. Example Let s be a bivariate symmetric polynomial and d ∈ N . Consider G = ( x d 1 + s ( x 2 , x 3 ) , x d 2 + s ( x 1 , x 3 ) , x d 3 + s ( x 1 , x 2 )). Then G is equivariant by the action of the symmetric group Sym 3 . Now, a set of the form { x ∈ R n : g i ( x ) ≥ 0 ∀ g ∈ G } is clearly Sym n -invariant, but does not directly fit into the pervious setup. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 6 / 17

Main geometric result We consider F = ( f 1 , . . . , f k ) and G = ( g 1 , . . . , g n ) sequences of polynomials in R [ x 1 , . . . , x n ]. Further, let d be the maximum of deg( f i ) and deg( g j ) for 1 ≤ i ≤ k and 1 ≤ j ≤ n . Denote by S ( F , G ) the semi-algebraic set defined by f 1 = · · · = f k = 0 , g 1 ≥ 0 , . . . , g n ≥ 0. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 7 / 17

Main geometric result We consider F = ( f 1 , . . . , f k ) and G = ( g 1 , . . . , g n ) sequences of polynomials in R [ x 1 , . . . , x n ]. Further, let d be the maximum of deg( f i ) and deg( g j ) for 1 ≤ i ≤ k and 1 ≤ j ≤ n . Denote by S ( F , G ) the semi-algebraic set defined by f 1 = · · · = f k = 0 , g 1 ≥ 0 , . . . , g n ≥ 0. Theorem Assume that for 1 ≤ i ≤ k, f i is Sym n invariant, that G is Sym n -equivariant and that deg( g j ) ≥ 2 for 1 ≤ j ≤ n. Then, the basic semi-algebraic set S ( F , G ) is empty if and only if S ( F , G ) contains a point contains a point with at most 2 d − 1 distinct coordinates. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 7 / 17

Ideas of the proof Deciding emptiness The proof uses three main ingredients: The fact that every symmetric polynomial can be uniquely written in terms of Newton sums ( p i := X i 1 + X i 2 + . . . + X i n ), a consequence of a representation result by Shchwartsman, an optimization problem. C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 8 / 17

Ideas of the proof Deciding emptiness The proof uses three main ingredients: The fact that every symmetric polynomial can be uniquely written in terms of Newton sums ( p i := X i 1 + X i 2 + . . . + X i n ), a consequence of a representation result by Shchwartsman, an optimization problem. Proposition (Shchwartsman) Let G be a sequence of equivariant polynomialslet deg g i ≤ d. Then d � s j · x j g i = i , j =0 where s j ∈ R [ x 1 , . . . , x n ] Sym n is symmetric and of degree ≤ d − j + 1 . C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 8 / 17

Ideas of the proof Suppose that S := S ( F , G ) is not empty and let y ∈ S . We set p 1 ( y ) = γ 1 , . . . , p d ( y ) = γ d and define N γ := { x ∈ R n : p 1 ( x ) = γ 1 , . . . , p d ( x ) = γ d } . C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 9 / 17

Ideas of the proof Suppose that S := S ( F , G ) is not empty and let y ∈ S . We set p 1 ( y ) = γ 1 , . . . , p d ( y ) = γ d and define N γ := { x ∈ R n : p 1 ( x ) = γ 1 , . . . , p d ( x ) = γ d } . Proposition Every symmetric polynomial f of degree ≤ d is constant on N γ . C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 9 / 17

Ideas of the proof Suppose that S := S ( F , G ) is not empty and let y ∈ S . We set p 1 ( y ) = γ 1 , . . . , p d ( y ) = γ d and define N γ := { x ∈ R n : p 1 ( x ) = γ 1 , . . . , p d ( x ) = γ d } . Proposition Every symmetric polynomial f of degree ≤ d is constant on N γ . Lemma Let d ≤ n, γ ∈ R d . Consider ( g 1 , . . . , g n ) a sequence of polynomials of degree at most d in R [ x 1 , . . . , x n ] which are Sym n - equivariant and ξ = ( ξ 1 , . . . , ξ n ) ∈ N γ . Then, there exist { α 1 , . . . , α t } ⊂ R with t ≤ d − 1 such that g i ( ξ ) = 0 if and only if ξ i ∈ { α 1 , . . . , α t } . C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 9 / 17

Main step of the proof Consider S ′ := S ∩ N γ . Then, S ′ is closed and bounded. Thus there exists a maximizer ξ = ( ξ 1 , . . . , ξ n ) for the function p d +1 on S ′ . If g i ( ξ ) > 0 for all i , then ∇ p d +1 ( ξ ) = � d k =1 λ k ∇ p k ( ξ ) (Lagrange) and thus d λ k ( k − 1) ξ ( � ( d + 1) ξ d i = i k − 1) , k =1 and thus |{ ξ 1 , . . . , x n }| ≤ d . C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 10 / 17

Main step of the proof Consider S ′ := S ∩ N γ . Then, S ′ is closed and bounded. Thus there exists a maximizer ξ = ( ξ 1 , . . . , ξ n ) for the function p d +1 on S ′ . If g i ( ξ ) > 0 for all i , then ∇ p d +1 ( ξ ) = � d k =1 λ k ∇ p k ( ξ ) (Lagrange) and thus d λ k ( k − 1) ξ ( � ( d + 1) ξ d i = i k − 1) , k =1 and thus |{ ξ 1 , . . . , x n }| ≤ d . If g i ( ξ ) = 0 for one i use the Lemma above to conclude that there are at most d − 1 choices for ξ i . C. Riener, M. Safey El Din Equivariant Semi-Algebraic Systems July 9, 2018 10 / 17