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A new method for reducing algebraic programs to polynomial programs

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Pith's one-line read Optimization problems with radical expressions can be rewritten as polynomial programs using one new variable per algebraic function plus isolating inequalities, and the tested benchmarks solve up to 50x faster.

desk verdict A useful, honest paper with a real algorithmic idea and a clearly stated but load-bearing density condition; worth refereeing carefully, not desk-rejecting. read the letter →

arxiv 2502.08210 v1 pith:CB4F3YLT submitted 2025-02-12 math.OC cs.SCmath.AG

Muhammad Maaz , Adam W. Strzeboński This is my paper · ORCID
classification math.OCcs.SCmath.AG MSC 90C2690C2214P10
keywords algebraicprogramspolynomialradicalexpressionsdefiningresultantThomencodingsum-of-squaresrelaxationcylindricaldecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to make algebraic programs—optimization or feasibility problems whose objective and constraints are algebraic functions built from radicals—tractable with the machinery of polynomial optimization. Its central proposal is a reformulation that introduces one new variable for each algebraic function appearing in the program, rather than one new variable for each distinct radical. The transformation has two algorithmic steps: construct a defining polynomial for the radical expression, then generate polynomial inequalities that single out the intended branch of that polynomial using derivative signs. The paper proves the construction correct and reports that, on two benchmark problems with added algebraic terms, the reformulation yields values equivalent to the standard one-variable-per-radical approach while solving up to 50 times faster. The catch is a density condition: the reformulation is equivalent only when the feasible set is contained in the closure of its interior, so the method is not directly applicable to equality-constrained or lower-dimensional feasible sets.

What carries the argument

The argument is carried by a two-stage symbolic construction. First, a defining polynomial p(z,x) for a radical expression is built recursively by resultants: each arithmetic operation or integer radical on subexpressions is translated into a resultant of the corresponding defining polynomials, eliminating the auxiliary variables of the subexpressions. Second, the intended root is isolated by a Thom encoding: on each connected component of the set where p and its derivatives share no root, the signs of the derivatives of p at the algebraic function are constant, and those signs, together with p = 0, form polynomial inequalities that select exactly that branch. The reformulation replaces each radical expression by one new variable z plus these constraints, and the paper proves the equivalence of the original and reformulated programs under the condition that the feasible set is contained in the closure of its interior.

What would settle it

Take a one-dimensional algebraic program with feasible set D = {0}, such as minimizing x subject to x = 0 and \sqrt{x} \geq 0, and apply the two algorithms literally: the resultant defining A is nonzero only for x \neq 0, so Algorithm 2 returns no isolating component at x = 0, and the reformulated polynomial program becomes infeasible even though the original has the feasible point x = 0. Checking whether the reformulation's optimal value equals the original value on such a lower-dimensional feasible set would directly test the claimed equivalence.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is that any algebraic program written with radical expressions can be rewritten as a polynomial program with exactly one auxiliary variable per algebraic function, provided the feasible set is dense in its interior. The key is to replace each radical expression f(x) by a variable z together with a defining polynomial p(z,x) = 0 and a collection of polynomial inequalities that isolate the branch corresponding to f. The defining polynomial is assembled recursively from resultants of the defining polynomials of the expression's subterms, and the isolating inequalities come from sign conditions on the derivatives of p, which are constant on each connected component of the set where the relevant resultants do not vanish. Over such components, the sign pattern of the derivatives uniquely characterizes the root, so the polynomial constraints together describe exactly the graph of f. The paper shows on a modified Goldstein-Price and a modified Rosenbrock problem that this reformulation produces semidefinite relaxations with far fewer variables and, in the reported experiments, up to roughly 50 times faster total solution times than the standard reformulation, without weakening the lower bounds.

Load-bearing premise

The whole equivalence rests on the feasible set being contained in the closure of its interior, so that the points where the isolating polynomials are well-behaved are dense in the feasible set; equality-constrained or lower-dimensional feasible sets can break the reduction.

Editorial extensions

If this is right

  • Polynomial-programming tools—cylindrical algebraic decomposition, Positivstellensatz infeasibility certificates, and sum-of-squares relaxations—become available for algebraic programs, with the number of auxiliary variables governed by the number of algebraic functions rather than the number of radicals.
  • On the modified Goldstein-Price benchmark the new reformulation uses 6-8 times fewer semidefinite variables and runs up to about 40 times faster in total time, and on the modified Rosenbrock benchmark the speedup reaches about 50 times.
  • The reformulation returns the same lower bounds as the straightforward one-variable-per-radical version at matching sum-of-squares degrees, indicating the relaxation is not weakened.
  • The isolating-inequality construction gives an algorithmic way to generate branch cuts for algebraic functions, since each branch is described by polynomial equalities and inequalities.
  • The equivalence is conditional on the feasible set being contained in the closure of its interior; for equality-constrained or lower-dimensional feasible sets the paper states the reformulation may exclude the true optimum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • For algebraic programs whose feasible set is full-dimensional but has equality constraints, the density condition could be restored by a preliminary dimension-reduction step or by solving a perturbed problem and taking limits; the paper does not explore this.
  • The derivative-sign isolation method could double as a symbolic real-valuedness test: if no connected component of A yields a real value for the radical expression over a given domain, the expression is non-real there.
  • A natural stress test would be to feed the two algorithms a nested radical expression with several variables, such as \sqrt{x^2 + \sqrt{y^2+1}}, and measure the degree of the defining polynomial and the number of isolating components; the paper's examples give no data on how these scale with expression depth.
  • In multi-constraint programs where the same algebraic function recurs, the single-variable reformulation may also reduce the number of blocks in the semidefinite program, a separate effect from the reduction in scalar variables that the reported timings do not isolate.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method for converting algebraic programs (optimization/feasibility problems whose objectives and constraints are radical expressions) into polynomial programs. For each radical expression it introduces a single new variable z, computes a defining polynomial via iterated resultants (Algorithm 1), and derives polynomial inequalities that isolate the intended branch of the algebraic function using Thom's lemma and a sign-of-derivatives encoding (Algorithm 2). Section 4 then reformulates the original program by adding these polynomial equalities and inequalities, and claims equivalence when a density condition holds. The method is demonstrated on modified Goldstein-Price and Rosenbrock problems, where the new formulation uses substantially fewer variables in the sum-of-squares relaxation and reports speedups of up to about 50x.

Significance. If the reduction is valid on its intended domain, this is a useful contribution: it connects algebraic programs to the mature toolkit for polynomial programs (CAD, Positivstellensatz certificates, sum-of-squares relaxations) and it can reduce the number of auxiliary variables compared with the standard one-variable-per-radical reformulation. The algorithms are concrete and the benchmark examples are checkable by hand; the reported speedups, while on toy instances, are plausible and clearly reported. The paper is also honest about several limitations, including the density condition and the possible combinatorial explosion of connected components. No code or data artifacts are included, so the experimental section is the main evidence for the performance claims.

major comments (3)
  1. [Section 4, first paragraph] The central equivalence claim is conditional on the density of Aall ∩ D in D, but this condition is not checked and it can fail for very simple algebraic programs. For example, min √x subject to x = 0 has feasible set D = {0} and optimum 0, but the proposed reformulation introduces z^2 - x = 0, z > 0, x = 0, which is infeasible because z = 0 violates z > 0. Thus the reformulation does not merely lose a certificate of optimality; it can turn a feasible program into an infeasible one. The paper acknowledges this in the Discussion, but the abstract and the opening of Section 4 state the reduction without this qualification. I ask the authors to restate the main theorem with the density hypothesis explicitly, to state that the method is intended for feasible sets contained in the closure of their interiors (or, more generally, for cases where the density condition is verified), and to provide an automatic or algorithmic check of the density condition rather than only saying that CAD can in principle decide it.
  2. [Lemma 2] The claimed equality dRes,x' = dp,x' dq,x + dp,x dq,x' is false in general; the correct statement is an upper bound, since cancellations in the determinant of the Sylvester matrix can lower the degree. For instance, with p = (y+1)x + 1 and q = (y+1)x + 2, one has Res_x(p,q) = y+1, whose degree in y is 1, while the formula gives 1·1 + 1·1 = 2. The proof's 'there exists a permutation' argument shows only that a term of that degree can occur, not that cancellations cannot occur. This does not invalidate Theorem 2, whose statement already uses 'at most', but the lemma and its proof should be corrected to an inequality, and Theorem 2's proof should be adjusted accordingly.
  3. [Theorem 3 and Algorithm 2] Theorem 3 is the theoretical basis for the isolating inequalities, but its proof delegates the key existence and continuity of the roots r1,...,rk to Lemma 3.6 of [Strzeboński, 2000] without stating that lemma or verifying its hypotheses in detail. Given that Algorithm 2's correctness depends on the assertion that the derivative sign pattern identifies a single continuous root on each connected component of A, the authors should either state and prove the needed lemma in the paper or give a fully self-contained argument, including a precise statement of how the real-valuedness of the radical expression is used.
minor comments (5)
  1. [Example 6] The component descriptions contain the algebraic number √5 (e.g., y > (7 + 3√5)x/2). Since the goal is a polynomial program over the reals, these constants should be encoded with an auxiliary variable and polynomial equations (e.g., w^2 = 5, w > 0) or the paper should state that algebraic constants are allowed.
  2. [Section 4.2] The sentence 'the second constraint can be eliminated as we cancel the z from the first constraint' is misleading at z = 0, where division by z is not valid. The subsequent inequality z^2 + 2 - x - y ≥ 0 is valid, but the derivation should be explained without dividing by z or with an explicit case discussion.
  3. [Tables 1 and 2] It would be helpful to specify whether the reported total times include the preprocessing overhead (0.135 s and 0.52 s stated in the text) or only the solver time, so that the speedup comparison is unambiguous.
  4. [Throughout] There are several typos and notational slips: 'Sylvster' in Definition 1, 'straightforwrad' in Section 4.3, 'This are the cases' in Section 4, and inconsistent use of x1/2 versus √x. A careful proofreading pass is needed.
  5. [Theorem 4] The bound on the number of connected components is quoted from [Basu et al., 1996] without specifying which parameter corresponds to the number of polynomials or the degree bound e; please make the application of that result explicit so the reader can verify the (d choose n)(O(e))^n expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reduction is built on standard real-algebraic facts, and the stated density condition is a genuine limitation rather than a disguised assumption of the result.

full rationale

The derivation chain is self-contained once the density condition is accepted. Algorithm 1 constructs defining polynomials by recursive resultants using Lemma 1 from Zippel, an external textbook result, and Algorithm 2 relies on Thom's Lemma from Basu et al. and on a continuity lemma from the author's earlier work on strict polynomial inequalities, which is a parameter-free building block rather than the target reformulation. The reformulation in Section 4 does not fit any parameter to the benchmarks and does not rename a fitted quantity as a prediction; it states a sufficient topological condition (D contained in the closure of its interior) for Aall ∩ D to be dense in D, and explicitly acknowledges that when this condition fails the method may not apply. The self-citations to Strzeboński 1999 (baseline reformulation), 2000 (root continuity), 2017 (CAD components), and Akritas and Strzeboński 2005 (root isolation) are to prior algorithmic tools whose assumptions do not include the equivalence claimed here. The resulting polynomial programs are genuinely different objects, with a single variable plus defining polynomial and sign conditions, rather than a restatement of the original radicals. Thus no step reduces by construction to its own input or to a fitted value, and no load-bearing premise is justified only by an unverified self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The method introduces no new parameters or entities. It relies on standard results from real algebraic geometry and on the structural assumption that the feasible set is dense in Aall, which limits applicability to full-dimensional programs.

assumptions (6)
  • standard math The resultant Res_x(p,q) vanishes iff p and q have a common root over the algebraic closure.
    Used throughout Algorithm 1 and in the definition of A in Section 3.
  • standard math For algebraic functions f,g with defining polynomials p,q, the polynomials in Lemma 1 are defining polynomials of f+g, f-g, f*g, f/g, and root expressions.
    Quoted from Zippel (1993) and used as the recursion step in Algorithm 1; it is a known result in computer algebra.
  • standard math Thom's lemma: the signs of a univariate polynomial and its derivatives separate its real roots.
    Used in Theorem 3 and Algorithm 2 to isolate a branch by derivative sign conditions.
  • standard math On a connected component of A, the roots of p(z,a) vary continuously and the sign of each derivative at each root is constant.
    Stated in Theorem 3 and used to justify evaluating signs at a single sample point.
  • domain assumption If Aall is open and dense in R^n and D is contained in the closure of its interior, then Aall ∩ D is dense in D, so the infimum of the objective over D equals that over Aall ∩ D.
    This density condition, stated in Section 4, is necessary for the reformulation to preserve the optimum; it holds for unconstrained, strict-inequality, and full-dimensional constraint sets, but cannot be assumed for equality-constrained problems.
  • domain assumption Connected components of the semialgebraic set A can be computed effectively, for example with cylindrical algebraic decomposition or other algorithms.
    Algorithm 2 requires enumerating components or a decomposition into connected sets; the paper acknowledges this is costly and may be prohibitive.

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Pith. "Pith review of A new method for reducing algebraic programs to polynomial programs." pith.science (2026). https://pith.science/paper/CB4F3YLT

@misc{pith2026250208210,
  author       = {Pith},
  title        = {Pith review of: A new method for reducing algebraic programs to polynomial programs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CB4F3YLT}},
  note         = {Machine review of arXiv:2502.08210}
}
read the original abstract

We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are a rich set of functions that includes polynomials themselves, but also ratios and radicals, and finite compositions thereof. When an algebraic program is given in terms of radical expressions, a straightforward way of reformulating into a polynomial program is to introduce a new variable for each distinct radical that appears. Hence, the rich theory and algorithms for polynomial programs, including satisfiability via cylindrical algebraic decomposition, infeasibility certificates via Positivstellensatz theorems, and optimization with sum-of-squares programming directly apply to algebraic programs. We propose a different reformulation, that in many cases introduces significantly fewer new variables, and thus produces polynomial programs that are easier to solve. First, we exhibit an algorithm that finds a defining polynomial of an algebraic function given as a radical expression. As a polynomial does not in general define a unique algebraic function, additional constraints need to be added that isolate the algebraic function from others defined by the same polynomial. Using results from real algebraic geometry, we develop an algorithm that generates polynomial inequalities that isolate an algebraic function. This allows us to reformulate an algebraic program into a polynomial one, by introducing only a single new variable for each algebraic function. On modified versions of classic optimization benchmarks with added algebraic terms, our formulation achieves speedups of up to 50x compared to the straightforward reformulation.

Figures

Figures reproduced from arXiv: 2502.08210 by the authors.

Figure 1
Figure 1. The polynomial x 2 −2xy +y 2 −2xz2 −2yz2 +z 4 implicitly defines four algebraic functions as its roots: ± √ x ± √y. Theorem 4. Suppose we have an algebraic function f(x1, . . . xn), with a defining polynomial p(z, x1, . . . , xn), which has degree in z of d, and the total degree of each monomial in the resultants that define A is bounded by e. Then, the algebraic function f exists over at most [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. The algebraic function √ x+ √y (red) spans four connected compo￾nents on the zero locus of its minimal polynomial x 2−2xy+y 2−2xz2−2yz2+z 4 (gray), each defined by polynomial inequalities (yellow). algorithm. In a more general setting, one would need to check whether Aall ∩ D is dense in D, which in principle can be done using the cylindrical algebraic decomposition algorithm. We may obtain multiple child problems, … view at source ↗

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Works this paper leans on

37 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    Mémoire sur les équations algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré

    Niels Henrik Abel. Mémoire sur les équations algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré. In Ludwig Sylow and Sophus Lie, editors, Œuvres Complètes de Niels Henrik Abel, volume I, pages 28--33. Grøndahl & Søn, 2nd edition, 1881 a . URL https://archive.org/details/oeuvrescompltes01abelgoog. [Firs...

    work page

  2. [2]

    Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré

    Niels Henrik Abel. Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré. In Ludwig Sylow and Sophus Lie, editors, Œuvres Complètes de Niels Henrik Abel, volume I, pages 66--87. Grøndahl & Søn, 2nd edition, 1881 b . URL https://archive.org/details/oeuvrescompltes01abelgoog. [First published 1826]

    work page

  3. [3]

    Akritas and Adam W

    Alkiviadis G. Akritas and Adam W. Strzebo \'n ski. A comparative study of two real root isolation methods. Nonlinear Analysis: Modelling and Control, 10 0 (4): 0 297--304, 2005. doi:10.15388/NA.2005.10.4.15110. URL https://doi.org/10.15388/NA.2005.10.4.15110

    work page doi:10.15388/na.2005.10.4.15110 2005

  4. [4]

    U ber die zerlegung definiter funktionen in quadrate. Abhandlungen aus dem Mathematischen Seminar der Universit \

    Emil Artin. \"U ber die zerlegung definiter funktionen in quadrate. Abhandlungen aus dem Mathematischen Seminar der Universit \"a t Hamburg , 5 0 (1): 0 100--115, 1927. doi:10.1007/BF02952513

    doi:10.1007/bf02952513 1927

  5. [5]

    S. Basu, R. Pollack, and M.F. Roy. Computing roadmaps of semi-algebraic sets on a variety. Journal of the AMS, 3: 0 55--82, 1999

    work page 1999

  6. [6]

    On the number of cells defined by a family of polynomials on a variety

    Saugata Basu, Richard Pollak, and Marie-Fran c oise Roy. On the number of cells defined by a family of polynomials on a variety. Mathematika, 43 0 (1): 0 120--126, 1996

    work page 1996

  7. [7]

    Algorithms in Real Algebraic Geometry

    Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in Real Algebraic Geometry. Springer Berlin, Heidelberg, 2006

    work page 2006

  8. [8]

    Semidefinite optimization and convex algebraic geometry

    Grigoriy Blekherman, Pablo A Parrilo, and Rekha R Thomas. Semidefinite optimization and convex algebraic geometry. SIAM, 2012

    2012

Show all 37 references
  1. [9]

    A tutorial on geometric programming

    Stephen Boyd, Seung-Jean Kim, Lieven Vandenberghe, and Arash Hassibi. A tutorial on geometric programming. Optimization and engineering, 8: 0 67--127, 2007

    2007

  2. [10]

    John F. Canny. The Complexity of Robot Motion Planning. MIT Press, Cambridge, MA, USA, 1988. ISBN 0-262-03136-1

    1988

  3. [11]

    Quantifier elimination for real closed fields by cylindrical algebraic decomposition--preliminary report

    George E Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition--preliminary report. ACM SIGSAM Bulletin, 8 0 (3): 0 80--90, 1974

    1974

  4. [12]

    Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets

    Michel Coste and Marie-Fran c oise Roy. Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. Journal of Symbolic Computation, 5 0 (1-2): 0 121--129, 1988

    1988

  5. [13]

    Z3: An efficient smt solver

    Leonardo De Moura and Nikolaj Bj rner. Z3: An efficient smt solver. In International conference on Tools and Algorithms for the Construction and Analysis of Systems, pages 337--340. Springer, 2008

    2008

  6. [14]

    Ueber die vollen invariantensysteme

    David Hilbert. Ueber die vollen invariantensysteme. Mathematische Annalen, 42 0 (3): 0 313--373, 1893. doi:10.1007/BF01444162

    doi

  7. [15]

    Global optimization of rational functions: a semidefinite programming approach

    Dorina Jibetean and Etienne de Klerk. Global optimization of rational functions: a semidefinite programming approach. Mathematical Programming, 106: 0 93--109, 2006

    2006

  8. [16]

    Solving non-linear arithmetic

    Dejan Jovanovi \'c and Leonardo De Moura. Solving non-linear arithmetic. ACM Communications in Computer Algebra, 46 0 (3/4): 0 104--105, 2013

    2013

  9. [17]

    Algebraic Functions, pages 119--134

    Konrad Knopp. Algebraic Functions, pages 119--134. Dover, New York, 1996

    1996

  10. [18]

    J. L. Krivine. Anneaux pr\' e ordonn\' e s. Journal d'Analyse Math\' e matique , 12: 0 307--326, 1964. doi:10.1007/bf02807438

    1964 doi

  11. [19]

    Semidefinite programming for min--max problems and games

    Rida Laraki and Jean B Lasserre. Semidefinite programming for min--max problems and games. Mathematical programming, 131: 0 305--332, 2012

    2012

  12. [20]

    Global optimization with polynomials and the problem of moments

    Jean B Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on optimization, 11 0 (3): 0 796--817, 2001

    2001

  13. [21]

    Semidefinite programming relaxations for semialgebraic problems

    Pablo A Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical programming, 96: 0 293--320, 2003

    2003

  14. [22]

    Positive polynomials on compact semi-algebraic sets

    Mihai Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42 0 (3): 0 969--984, 1993. doi:10.1512/iumj.1993.42.42045

    1993 doi

  15. [23]

    Riflessioni intorno alla soluzione delle equazioni algebraiche generali opuscolo del cav

    Paolo Ruffini. Riflessioni intorno alla soluzione delle equazioni algebraiche generali opuscolo del cav. dott. Paolo Ruffini. Societa Tipografica, 1813

  16. [24]

    PICOS : A Python interface to conic optimization solvers

    Guillaume Sagnol and Maximilian Stahlberg. PICOS : A Python interface to conic optimization solvers. Journal of Open Source Software, 7 0 (70): 0 3915, February 2022. ISSN 2475-9066. doi:10.21105/joss.03915

    2022 doi

  17. [25]

    The k-moment problem for compact semi-algebraic sets

    Konrad Schm \"u dgen. The k-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289 0 (1): 0 203--206, 1991. ISSN 0025-5831. doi:10.1007/bf01446568

    1991 doi

  18. [26]

    A nullstellensatz and a positivstellensatz in semialgebraic geometry

    Gilbert Stengle. A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207: 0 87--97, 1974

    1974

  19. [27]

    Strzebo \'n ski

    Adam W. Strzebo \'n ski. Global optimization of real algebraic functions subject to algebraic equation and inequality constraints. https://www.researchgate.net/publication/270904844, 1999

    1999

  20. [28]

    Strzebo \'n ski

    Adam W. Strzebo \'n ski. Solving systems of strict polynomial inequalities. Journal of Symbolic Computation, 29: 0 471--480, 2000

    2000

  21. [29]

    Strzebo \'n ski

    Adam W. Strzebo \'n ski. Cad adjacency computation using validated numerics. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC 2017, pages 413--420. ACM, 2017

    2017

  22. [30]

    Surjanovic and D

    S. Surjanovic and D. Bingham. Virtual library of simulation experiments: Test functions and datasets. Retrieved November 8, 2024, from http://www.sfu.ca/ ssurjano, 2013

    2024

  23. [31]

    A decision method for elementary algebra and geometry

    Alfred Tarski. A decision method for elementary algebra and geometry. In Quantifier elimination and cylindrical algebraic decomposition, pages 24--84. Springer, 1951

    1951

  24. [32]

    B. L. van der Waerden. Modern Algebra, Volume II. Springer, 1931

    1931

  25. [33]

    Geometric optimization and sums of algebraic functions

    Antoine Vigneron. Geometric optimization and sums of algebraic functions. ACM Transactions on Algorithms (TALG), 10 0 (1): 0 1--20, 2014

    2014

  26. [34]

    SumOfSquares.py , 2024

    Chenyang Yuan. SumOfSquares.py , 2024. URL https://github.com/yuanchenyang/SumOfSquares.py

    2024

  27. [35]

    Geometric programming

    CE Zener, RJ Duffin, and EL Peterson. Geometric programming. John Wiley, & Sons, Inc., New York, 1967

    1967

  28. [36]

    Effective Polynomial Computation

    Richard Zippel. Effective Polynomial Computation. Kluwer Academic Press, Boston, MA, 1993

    1993

  29. [37]

    Zero testing of algebraic functions

    Richard Zippel. Zero testing of algebraic functions. Information processing letters, 61 0 (2): 0 63--67, 1997

    1997

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