By Alt R., Vignes J.

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171–188, 1997. 31 30. , Hou, X. R. & Zeng, Z. , A Complete Discrimination System for Polynomials, Science in China, Series E 39:6, 628–646, 1996. 31, 33, 34 31. Yang, L. & Xia, B. , Explicit Criterion to Determine the Number of Positive Roots of a Polynomial, MM Research Preprints, No. 15, Beijing, pp. 134–145, 1997. 31 32. , Zhang, J. Z. & Hou, X. , A Criterion of Dependency between Algebraic Equations and Its Applications, Proc. , Beijing, pp. 110–134, 1992. 31 33. , Zhang, J. Z. & Hou, X. , An Eﬃcient Decomposition Algorithm for Geometry Theorem Proving Without Factorization, Proc.

After this simpliﬁcation only the package qerrc can eliminate the remaining quantiﬁer, but the obtained elimination result ϕ is very large. Though it has only 81 atomic formulas a textual representation contains approximately 500 000 characters. However, we have found a quantiﬁer-free description of the image of the function describing the Enneper surface. One of the atomic formulas contained in the result is the equation p = 0. But this fact does not imply any direction of the equivalence we want to prove.

2; P1 (g1 , g2 , . . , gt ) = {h(u)|h ∈ U1 }, where Ui means the set consisting of all the polynomials in each GDL(fi , q) with q belonging to mset(Pi+1 ). Analogously, we can deﬁne P1 (g1 , . . , gj ) (1 ≤ j ≤ t). 1. The necessary and suﬃcient condition for system T S to have a given number of distinct real solution(s) can be expressed in terms of the signs of the polynomials in P1 (g1 , g2 , . . , gt ). Proof. First of all, we regard fs and every gi as polynomials in xs . 2 we know that under constraints gi ≥ 0, 1 ≤ i ≤ t, the number of distinct real solutions of fs = 0 can be determined by the signs of polynomials in Ps ; let hj (1 ≤ j ≤ l) be the polynomials in Ps ; we regard every hj and fs−1 as polynomials in xs−1 , repeat the same argument as what we did for fs and gi ’s, then we get that, under constraints gi ≥ 0, 1 ≤ i ≤ t, the number of distinct real solutions of fs = 0, fs−1 = 0 can be determined by the signs of polynomials in Ps−1 ; do the same argument until P1 (g1 , g2 , .

### 10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics by Alt R., Vignes J.

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