A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

On the Betti numbers of semialgebraic sets defined by few quadratic inequalities

Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We ...

Computing the Top Betti Numbers of Semi-algebraic Sets Defined by Quadratic Inequalities in Polynomial Time

For any l > 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≤ 0, . . . , Ps ≤ 0, where each Pi ∈ R[X1, . . . ,Xk ] has degree ≤ 2, and computes the top l Betti numbers of S, bk−1(S), . . . , bk−l(S), in polynomial time. The complexity of the algorithm, stated more precisely, is ∑l+2 i=0 ( s i ) k2 O(min(l,s)) . For fixed l, the complexity of the algorithm ...

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S ⊂ Rk+m be a compact semi-algebraic set defined by P1 ≥ 0, . . . , P` ≥ 0, where Pi ∈ R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ `. Let π denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of π(S) is bounded by (k +m). We also present an algorithm for computing the the first q Betti numbers of π(S), whose com...

Computing the Betti Numbers of Semi-algebraic Sets Defined by Partly Quadratic Systems of Polynomials

Let R be a real closed field, Q ⊂ R[Y1, . . . , Y`, X1, . . . , Xk], with degY (Q) ≤ 2, degX(Q) ≤ d,Q ∈ Q,#(Q) = m, and P ⊂ R[X1, . . . , Xk] with degX(P ) ≤ d, P ∈ P,#(P) = s. Let S ⊂ R`+k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We describe an algorithm for computing the the Betti numbers of S generalizing a similar alg...