Faster real root decision algorithm for symmetric polynomials

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by taking advantage of the symmetry invariance property. The complexity of our algorithm is polynomial in $d^s, {{n+d} \choose d}$, and ${{n} \choose {s+1}}$, where $n$ is the number of variables and $d$ is the maximal degree of $s$ input polynomials defining the real algebraic set under study. In particular, this complexity is polynomial in $n$ when $d$ and $s$ are fixed and is equal to $n^{O(1)}2^n$ when $d=n$.


INTRODUCTION
Let = ( 1 , . . ., ) be polynomials in the multivariate polynomial ring Q[ 1 , . . ., ] and let ( ) ⊂ C be the algebraic set defined by .We denote by R ( ) := ( ) ∩ R the set of solutions in R to the system .In addition we assume that all 's are invariant under the action of the symmetric group , that is, are symmetric polynomials (or equivalently, -invariant polynomials).
Under this invariance property, we design an algorithm which, on input , decides whether R ( ) is empty or not.As is typical for such problems, we assume that the Jacobian matrix of with respect to 1 , . . ., has rank at any point of ( ).In this case the Jacobian criterion [22,Thm 16.19] implies that the complex algebraic set ( ) is smooth and ( − )-equidimensional (or empty).
Previous work.The real root decision problem for polynomial systems of equations (and more generally systems of inequalities) lies at the foundations of computational real algebraic geometry.Algorithms for solving polynomial systems over the real numbers start with Fourier [29] who provided a first algorithm for solving linear systems of inequalities (rediscovered in 1919 by Dines [21]).
These algorithms are important because they make the first connection with elimination theory.Tarski's theorem [54] states that the projection of a semi-algebraic set on a coordinate subspace is a semi-algebraic set.This theorem, and its algorithmic counterpart which relies on Sturm's theorem for real root counting in the univariate case, enable recursive algorithmic patterns (eliminating variables one after another).The first algorithm with an elementary recursive complexity, Cylindrical Algebraic Decomposition, is due to Collins (see [19] and references in [16,17,24,35,37,38,51,52] for various further improvements).
It turns out that these algorithms run in time doubly exponential in [13,20].Note that some variants actually solve the quantifier elimination problem, a much more general and difficult computational problem than the real root decision problem.
Algorithms which solve the real root decision problem in time singly exponential in and polynomial in the maximum degree of the input were pioneered by Grigoriev and Vorobjov [32] and Renegar [40], and further improved by Canny [15], Heintz, Roy and Solernó [34] and Basu, Pollack and Roy [8].The method used in this framework is referred to as the critical point method.It reduces the real root decision problem to the computation of finitely many complex critical points of a polynomial map which reaches extrema at each connected component of the semi-algebraic set under study.
The algorithm proposed here for solving the real root decision problem for systems of symmetric polynomial equations also builds on the critical point method.It borrows ideas from probabilistic algorithms which have been designed to obtain sharper complexity estimates (e.g.cubic either in some Bézout bound attached to some critical point system or in some geometric intrinsic degree) and obtain practical performances that reflect the complexity gains [2][3][4][5][6][7]45].These algorithms make use of geometric resolution or symbolic homotopy techniques to control the complexity of the algebraic elimination step (see e.g.[31,46] and references therein), and of regularity assumptions to easily derive critical point systems from the input polynomials.
Under the Jacobian criterion assumptions, critical points are defined as the intersection of the affine variety ( ) with a determinantal variety derived from a certain Jacobian matrix.The design of dedicated algebraic elimination algorithms for this particular setting has attracted some attention already [1,27,33,47,50].
When adding the symmetry property to polynomials defining the variety and the polynomial map for which one computes the critical points, significant improvements have been achieved recently in [25] by using the symbolic homotopy algorithms in [36].
These improvements, which allows one to obtain complexity gains related to the combinatorial complexity of the symmetric group, also borrow ideas from algebraic algorithms working with data which are invariant by the action of this group [28].We emphasize that taking advantage of symmetries in data is a topical and difficult issue, which involves a variety of methodologies [14,18,26,39,53].
In [55], Timofte proves a breakthrough result which is now known as the degree principle.It states that a symmetric polynomial of degree with real coefficients has real solutions if and only if one of these solutions has at most distinct coordinates.
This shows that when is fixed and grows, the real root decision problem can be solved in polynomial time.This is far better than computing at least one sample point per connected component (see also [10][11][12]), and is one of the rare interesting cases where the best known algorithms for these two problems admit different complexities.This is also the starting point of several results which enhance the real root decision problem and polynomial optimization under some -invariance property for classes of problems where remains fixed and grows (see [30,41,42,44] and [43] for equivariant systems).
Main contributions.Being able to leverage -invariance for critical point computations is not sufficient to solve root decision problems more efficiently using the critical point method.Additional techniques are needed.
Indeed, to solve the real root decision problem by finding the critical points of a polynomial map , one typically defines as the distance from points on the variety to a generic point.This map reaches extrema at each connected component of the semialgebraic set under study.However, the map is not symmetric.If it was, our problem would be solved by the critical point algorithm of [25].Unfortunately there does not appear to be an obvious symmetric map that fits the bill.
Instead, our approach is to apply the critical point method on individual -orbits, with suitable found for each orbit.Thus while we cannot use the critical point algorithm of [25] directly we can make use of the various subroutines used in it to construct a fast decision procedure.Intuitively, working with -orbits is the same as separately searching for real points having distinct coordinates, or real points having two or more coordinates which are the same, or groups of coordinates each of which has equal coordinates and so on.In each case an orbit can be described by points having or fewer pairwise distinct coordinates, a key observation in constructing generic maps invariant for each orbit.T 1.1.Let = ( 1 , . . ., ) be symmetric polynomials in Q[ 1 , . . ., ] having maximal degree .Assume that the Jacobian matrix of with respect to 1 , . . ., has rank at any point of ( ).Then there is a Monte Carlo algorithm Real_emptiness which solves the real root decision problem for with operations in Q.Here the notion ˜indicates that polylogarithmic factors are omitted.
The remainder of the paper proceeds as follows.The next section reviews known material, on invariant polynomials over products of symmetric groups, the tools we use to work with -orbits, and our data structures.Section 3 discusses our smoothness requirement and shows that it is preserved by alternate representations of invariant polynomials.Section 4 shows how we construct critical point functions along with their critical point set.This is followed in Section 5 by a description of our algorithm along with a proof of correctness and complexity.The paper ends with a section on topics for future research.
In this case, we have

Describing -orbits via Partitions
-orbits are subsets of C that play a central role in our algorithm.In this section, we review notation and description of -orbits, along with the form of the output used in [25].
A simple way to parameterize -orbits is through the use of partitions of .A sequence = ( 1 1 . . .), where 1 < • • • < and 's and 's are positive integers, is called a partition of if The length of the partition is defined as For a partition = ( 1 1 . . . ) of , we use the notation from [25, Section 2.3] and let denote the set of all points in C that can be written as For any point in C , we define its type as the unique partition of such that there exists ∈ such that ( ) ∈ , with the , 's in (1) pairwise distinct.Points of a given type = ( 1 1 . . . ) are stabilized by the action of := 1 ×• • •× , the cartesian product of symmetric groups .

Zero-Dimensional Parametrizations
The subroutines we use from [25] give their output in terms of zero-dimensional parametrizations, which are defined as follows.

Let
⊂ C be a variety of dimension zero, defined over where is a new indeterminate, and deg( a linear form in variables such that ( 1 , . . ., ) = ′ (so the roots of are the values taken by on ).
When these conditions hold, we write = (R).Representing the points of by means of rational functions with ′ as denominator is not necessary, but allows for a sharp control of the bit-size of the output.

PRESERVING SMOOTHNESS
In our main algorithm, we assume that our input system = ( 1 , . . ., ) satisfies the following smoothness condition (A) : the Jacobian matrix of has rank at any point of ( ).
In this section, we discuss consequences of this assumption for symmetric polynomials.

P
. For any polynomial in Q[ 1 , . . ., ], applying the operator T on evaluates at = , for 1 ≤ ≤ , 1 ≤ ≤ and in , .By the multivariable chain rule, If is symmetric, for , ′ in , , we then have This argument can be extended to a sequence of polynomials to obtain our claim. .Then , and so This implies that Jac(T ( )) • is equal to ( , , , , , , ), with This is precisely T (Jac( )).
When we represent 1 × • • • × -invariant functions in terms of Newton sums, we can show that the new representation also preserves condition (A).
Similarly, instead of using a row-scaled Vandermonde matrix as in (4), we can use as the Jacobian matrix of elementary symmetric functions in .This gives a similar result but for the polynomials 1 , . . ., .

CRITICAL LOCI
If ⊂ C ℓ is an equidimensional algebraic set, and a polynomial function defined on , a non-singular point ∈ is called a critical point of on if the gradient of at is normal to the tangent space of at .If = ( 1 , . . ., ) are generators of the ideal associated to , then is the right kernel of the Jacobian matrix Jac( ) of evaluated at .In the cases we will consider, this matrix will have rank at all points of (that is, satisfies condition A).The set of critical points of the restriction of to is then defined by the vanishing of , and of the ( + 1)-minors of the Jacobian matrix Jac( , ) of and .

Finiteness through genericity
-invariant mappings and discuss the properties of their critical points on ( ) ⊂ C ℓ .
For 1 ≤ ≤ , let = ( 1, , . . ., , ) be new indeterminates, and recall that , is the -th Newton sum for the variables .Set where = 1 if is odd and = 0 if is even.So has even degree and is invariant under the action of
Let and be defined as in (5) and Lemma 2.2, respectively.For = 1, . . ., , set = +1, , and let ℎ 1 , . . ., ℎ = 1 , . . ., .In particular, Lemma 2.2 implies that is given by The sequence S can be rewritten as where is a multi-row-scaled Vandermonde matrix which is the Jacobian matrix of with respect to .This matrix has full rank at any point in the open set U defined in Subsection 4.1.
In particular, for any ∈ C 1 × • • • × C , the intersection of (S ) with C × U is contained in the preimage by the map Id × of the vanishing set of the sequence : ℎ 1 , . . ., ℎ , Since for all 1 ≤ ≤ , defines a map with finite fibers (by Newton identities and Vieta's formula, the preimage by of some point is the set of roots of some polynomial of degree ), we deduce that and consequently Id × define maps with finite fibers.Thus It remains to investigate finiteness properties of ( ).Using techniques from [23], one could give a simple exponential upper bound the degree of a hypersurface containing the complement of A. R → R be a real polynomial, where is the homogeneous component of degree of .Assume further that the leading form of is positive definite; then, is proper.In particular, the map

Finding extrema using proper maps
, with the Newton sums in 1 , . . ., and all in Q, is proper.We can extend this to blocks of variables.

MAIN RESULT
Let = ( 1 , . . ., ) be a sequence of symmetric polynomials in Q[ 1 , . . ., ] that satisfies condition (A).In this section we present an algorithm and its complexity to decide whether the real locus of ( ) is empty or not.
To exploit the symmetry of and to decide whether the set R ( ) is empty or not, our main idea is slicing the variety ( ) with hyperplanes which are encoded by a partition of .This way, we obtain a new polynomial system which is invariant under the action := 1 × • • • × of symmetric groups.We proved in Lemma 3.4 that this new system also satisfies condition (A).We then use the critical point method to decide whether the real locus of the algebraic variety defined by this new system is empty or not by taking a -invariant map as defined in the previous section.
Here = ( 1, , . . ., , ) denotes the vector of elementary symmetric polynomials in variables , with each , having degree for all , .L 5.1.Let , , and as above.Assume further that has finitely many critical points on ( ).Then there exists a randomized algorithm Critical_points ( , , ) which returns a zero-dimensional parametrization of the critical points of restricted to ( ).The algorithm uses The number of solutions is at most .

P
. The Critical_points procedure contains two steps: first finding and from and and then computing a representation for the set ( , ) of critical points of on ( ).The first step can be done using the algorithm Symmetric_Coordinates from [25, Lemma 9], which uses ˜ ℓ+ 2 operations in Q.
Since each , has degree , it is natural to assign a weight to the variable , , so that the polynomial ring Q[ 1 , . . ., ] is weighted of weights (1, . . ., 1 , . . ., 1, . . ., ).The weighted degrees of and are then equal to those of and , respectively.To compute a zero-dimensional parametrization for ( , ) we use the symbolic homotopy method for weighted domain given in [36,Thm 5.3] (see also [25,Sec 5.2] for a detailed complexity analysis).This procedure is randomized and requires ˜ 2 ( + 5 ) 4 Γ operations in Q.
Furthermore, results from [36,Thm 5.3] also imply that the number of points in the output is at most .Thus, the total complexity of the Critical_points algorithm is then ˜ 2 ( + 5 ) 4 Γ operations in Q.

The Decide procedure
Consider a partition = ( 1 1 . . . ) of , and let R = ( , 1,1 , . . ., 1 ,1 , . . ., 1, , . . ., , , ) be a parametrization which encodes a finite set ⊂ C ℓ .This set lies in the target space of the algebraic map : → C ℓ defined in Subsection 2.2 as where , ( 1, , . . ., , ) is the -th elementary symmetric function in 1, , . . ., , for = 1, . . ., and = 1, . . ., .Let be the preimage of by .In this subsection we present a procedure called Decide(R ) which takes as input R , and decides whether the set contains real points.In order to do this, a straightforward strategy consists in solving the polynomial system to invert the map .Because of the group action of 1 × • • • × , we would then obtain 1 !• • • ! points in the preimage of a single point in : we would lose the benefit of all that had been done before.
This difficulty can be bypassed by encoding one single point per orbit in the preimage of the points in .This can be done via the following steps.
(i) Group together the variables = ( 1, , . . ., , ) which encode the values taken by the elementary symmetric functions ,1 , . . ., , (see Sec. 2.2) and denote by ,1 , . . ., , the parametrizations corresponding to 1, , . . ., , ; (ii) Make a reduction to a bivariate polynomial system by considering the polynomial with coefficients in and "solving" the system = = 0.Here we recall that ∈ Q[ ] and is square-free, so that and ′ are coprime.(iii) It remains to decide whether, for all 1 ≤ ≤ , there is a real root of such that when replacing by in , the resulting polynomial has all its roots real.To do this we proceed by performing the following steps for 1 ≤ ≤ : (1) first we compute the Sturm-Habicht sequence associated to , in Q[ ] (the Sturm-Habicht sequence is a signed subresultant sequence, see [9, Chap.9, Algo.8.21]); (2) next, we compute Thom-encodings of the real roots of , which is a way to uniquely determine the roots of a univariate polynomial with real coefficients by means of the signs of its derivatives at the considered real root (see e.g.[9, Chap.10, Algo.10.14]); (3) finally, for each real root of , evaluate the signed subresultant sequence at [9, Chap.10, Algo.10.15] and compute the associated Cauchy index to deduce the number of real roots of (see [9,Cor. 9.5]).(iv) For a given real root of , it holds that, for all 1 ≤ ≤ , the number of real roots of equals its degree, if and only if is non-empty.
The above steps describe our Decide, which returns false if contains real points, else true.
For a partition , we first find the polynomials := T ( ), which are -invariant in Q[ 1 , . . ., ], where T is defined as in (2).By Corollary 3.4, satisfies condition (A), so we can apply the results of Section 4.
At the final step, we run the Decide(R ) in order to determine whether the preimage of by the map contains real points.
Algorithm 1 Real_emptiness( ) with < such that satisfies (A) Output: false if ( ) ∩ R is non-empty; true otherwise (1) for all partitions = ( 1 1 . . . ) of of length at least , do (a) compute = T ( ), where T is defined in (2) (b) using a chosen ∈ A, where A is defined as in Prop 4.1 , we construct as in ( 5) and then compute (c Assume that, on input symmetric as above, and satisfying condition (A), for all partitions of length at least , is chosen in A and that all calls to the randomized algorithm Criti-cal_points return the correct result.Then Algorithm Real_emptiness returns true if ( ) ∩ R is empty and otherwise it returns false.

P
. Since satisfies condition (A), Lemma 3.4 implies that also satisfies this condition.Then, by the Jacobian criterion [22,Thm 16.19], ( ) is smooth and equidimensional of dimension (ℓ − ), where ℓ is the length of .Therefore, if ℓ < , then the algebraic set ( ) is empty.Thus, the union of ( ) ∩ U where U is the open set defined in Subsection 4.1 and runs over the partitions of of length at least , forms a partition of ( ).Hence, ( ) ∩ R is non-empty if and only if there exists at least one such partition for which ( ) ∩ U ∩ R is non-empty.
We already observed that for all , does satisfy condition (A).Since we have assumed that each time Step 1b is performed, is chosen in A , we apply Proposition 4.4 to deduce that the conditions of Lemma 5.1 are satisfied.Hence, all calls to Critical_points are valid.
Note that since we assume that all these calls return the correct result, we deduce that their output encodes points which all lie in ( ).Hence, if ( ) ∩R is empty, applying the routine Decide on these outputs will always return true and, all in all, our algorithm returns true when ( ) ∩ R is empty.
It remains to prove that it returns false when ( ) ∩ R is nonempty.Note that there is a partition such that ( ) ∩ R is nonempty and has an empty intersection with the complement of U .That is, all connected components of ( ) ∩ R are in U .
Let be such a connected component.By Lemma 4.5, the map is proper, and non-negative.Hence, its restriction to ( ) ∩R reaches its extremum at all connected components of ( ) ∩ R .This implies that the restriction of to ( ) has real critical points which are contained in (and by Proposition 4.1 there are finitely many).Those critical points are then encoded by the output of the call to Critical_points (Step 1c) and false is returned.

Complexity analysis
Let = max(deg( )).First for a partition , applying T to takes linear time in ( + ), the number of monomials of and the cost of Step 1b is nothing.At the core of the algorithm, computing R at Step 1c requires ˜ 2 ( + 5 ) 4 Γ operations in Q by Lemma 5.1, where = max( , deg( )).Also, the degree of R is at most .
In order to determine the cost of the Decide process at Step 1d, let be the degree of and be the maximum of the partial degrees of 's w.r.t. .By the complexity analysis of [9, Algo.8.21 ; Sec.

8.3.6],
Step (1) above is performed within 4 arithmetic operations in Q[ ] using a classical evaluation interpolation scheme (there are polynomials to interpolate, all of them being of degree ≤ 2 ).Step (2) above requires 4 log( ) arithmetic operations in Q (see the complexity analysis of [9, Algo 10.14; Sec.10.4]).Finally, in Step (3), we evaluate the signs of polynomials of degree ≤ 2 at the real roots of (of degree ) whose Thom encodings were just computed.This is performed using 3 ((log( ) + )) arithmetic operations in Q following the complexity analysis of [9,Algo 10.15;Sec. 10.4].The sum of these estimates lies in 4 + 4 ((log( ) + )) .Now, recall that the degree of is the degree of R , so ≤ .The degree of w.r.t.equals and ≤ .This means ≤ .All in all, we deduce that the total cost of this final step lies in 4 + 2 , which is negligible compared to the previous costs.
In the worst case, one need to consider all the partitions of of length at least .Thus the total complexity of Real_emptiness is ,ℓ ≥ ˜ 2 ( + 5 ) 4 Γ operations in Q.In addition, Lemma 34 in [25] implies that + + + 1 operations in Q.
The output of our algorithm is consistent with the fact that the point (1, 1, 1/2, 1/2) is in R ( ).

TOPICS FOR FUTURE RESEARCH
Determining topological properties of a real variety R ( ) is an important algorithmic problem.Here we have presented an efficient algorithm to determine if R ( ) is empty or not.More generally, we expect that the ideas presented here may lead to algorithmic improvements also in more refined questions, like computing one point per connected component or the Euler characteristic for a real symmetric variety.Furthermore, while our complexity gains are significant for symmetric input we conjecture that we can do better in certain cases.In particular, when the degree of the polynomials is at most then we expect we that a combination with the topological properties of symmetric semi algebraic sets found in [12,Prop 9] can reduce the number of orbits considered, for example, instead of we might only need /2 for fixed .Finally, a generalization to general symmetric semi algebraic sets should be possible.
with each in C , we denote by the polynomials in C[ 1 , . . ., ] obtained by evaluating the indeterminates at in , for all .Further, we denote by U ⊂ C ℓ the open set consisting of points = ( 1 , . . ., ) such that the coordinates of are pairwise distinct for = 1, . . ., .Note that U depends on the partition = ( 1 1 . . .); when needed because of the use of different partitions, we will denote it by U .

L 4 . 2 .
with each in C , we denote by S the polynomials in C[ 1 , . . ., , 1 , . . . .] obtained by evaluating at in S , for all .Finally, denote by the projection from the ( , )-space C +ℓ to the -space C ℓ .Suppose that satisfies condition (A).Then for ∈ C 1 × • • • × C , ( (S )) is the critical locus of the restriction of the map to ( ).P. For any ∈ C 1 × • • • × C , we denote by ( , ) the set of critical points of the restriction of to ( ).Since satisfies condition (A), the set ( , ) is given by