Our work is divided into two parts. In the first part, we present structural results concerning the tropicalizations of semialgebraic sets. In the second part, we discuss algorithmic consequences of our approach.
In Chapter 3, we start by studying the tropicalizations of general semialgebraic sets. In the case of (complex) algebraic varieties, a fundamental result of Bieri and Groves [BG84, EKL06]
states that if (K,val) is an algebraically closed valued field with value group equal to R and S ⊂ (K∗)n is an algebraic variety in the torus, then val(S) ⊂ Rn is a union of polyhedra.
Alessandrini [Ale13] proved an analogue of this theorem for definable subsets of Hardy fields of polynomially bounded o-minimal structures. His analysis implies that if(K,val)is a real closed field with value group equal toRand convex valuation, and S ⊂(K∗)n is a semialgebraic set, thenval(S) is a union of polyhedra. The results of Alessandrini also apply to fields with value groups smaller than R. In Chapter 3, we give a constructive proof of this result. The proof is based on the Denef–Pas quantifier elimination [Pas89] and applies to fields with arbitrary value groups (not only the subgroups of R). Denef–Pas quantifier elimination also gives a transfer principle that is used later to prove the tropical analogue of the Helton–Nie conjecture for arbitrary real closed valued fields.
1.2. Our contribution 13 Theorem A (Theorem 3.1). Suppose that(K,val)is a real closed field equipped with a convex and nontrivial valuation val: K → Γ ∪ {−∞}. Let S ⊂ Kn be a semialgebraic set. Then, the set val(S)⊂(Γ ∪ {−∞})n is semilinear and has closed strata. Conversely, any semilinear subset of (Γ∪ {−∞})n that has closed strata arises as an image by valuation of a semialgebraic set.
Given a setS, it is sometimes possible to find a simple description of val(S). An important result in this direction is the Kapranov theorem [EKL06, Theorem 2.1.1], which states that if K is the field of Puiseux series with complex coefficients and S := {x∈ (K∗)n:P(x) = 0} is a hypersurface given by a polynomial P ∈K[X1, . . . , Xn], thenval(S) is fully described by the formal tropicalization of the polynomial P. This statement is not true for real semialgebraic states, even if they are described by one polynomial (in)equality. However, thanks to the theorem above, we can give an explicit description of the images by valuation of semialgebraic sets defined by systems of inequalities, provided that a certain regularity condition is satisfied.
Theorem B (Theorem 3.4). Let K denote the field of Puiseux series with real coefficients and let S ⊂Kn>0 be a semialgebraic set defined as
S :={x∈Kn>0:P1(x)□1 0, . . . ,Pm(x)□m0},
where Pi ∈K[X1, . . . , Xn] are nonzero polynomials and □∈ {⩾, >}m. Let Pi := trop(Pi) for all iand suppose that C⩾(P1, . . . , Pm) has regular support. Then
val(S) ={x∈Rn:∀i, Pi+(x)⩾Pi−(x)}.
The theorem above is a generalization of a result of Develin and Yu [DY07] who proved Theorem B in the case where polynomialsPiare affine. Furthermore, we note that the condition
“C⩾(P1, . . . , Pm) has regular support” is generic—it is satisfied if the coefficients of the tropical polynomialsP1, . . . , Pm lay outside of some set of measure zero.
We then turn our attention to the tropicalizations of convex semialgebraic sets defined over Puiseux series. In Chapter 4 we introduce the notion of tropical spectrahedra, which are sets defined as images by valuation of spectrahedra over Puiseux series. We show that a class of tropical spectrahedra defined by Metzler matrices can be described explicitly using polynomial inequalities of degree two in the tropical semifield. Subsequently, we use this class to give an explicit description of generic tropical spectrahedra. It turns out that only 2×2 minors are needed to describe a generic tropical spectrahedron. This extends the result of Yu [Yu15] who tropicalized the positive semidefinite cone.
Theorem C (Theorem 4.28). Suppose that Q(0), . . . ,Q(n) ∈ Km×m are symmetric matrices, letQ(x) :=Q(0)+x1Q(1)+· · ·+xnQ(n)andS :={x∈Kn⩾0:Q(x)≽0}be the associated spec-trahedron. Moreover, suppose that the matrices val(Q(0)), . . . ,val(Q(n)) ∈ Tm×m are generic.
Then, the set val(S) is described by tropical polynomial inequalities that are given by a tropical variant of the 2×2 minors of the affine pencil Q(x).
In Chapter 5, we study general convex semialgebraic sets and give multiple equivalent char-acterizations of their images by valuation. In particular, we show that the tropical analogue of the Helton–Nie conjecture is true.
Théorème A (Theorem 5.5). LetS⊂Tn. Then, the following conditions are equivalent:
(a) Sis a tropicalization of a convex semialgebraic set;
14 Chapter 1. Introduction (b) Sis tropically convex and has closed semilinear strata;
(c) Sis a projection of a tropical Metzler spectrahedron;
(d) there exists a projected spectrahedron S ⊂Kn⩾0 such thatval(S) =S.
This theorem is the first place where the connection with stochastic mean payoff games plays a role—one of the ingredients of the proof is a lemma of Zwick and Paterson [ZP96] that was originally used to prove a reduction from discounted mean payoff games to simple stochastic games. Using the transfer principle given by the Denef–Pas quantifier elimination, we can then show that the classical Helton–Nie conjecture is true “up to taking the valuation.”
Theorem D (Theorem 5.2). Let K be a real closed valued field equipped with a nontrivial and convex valuationval:K→Γ ∪ {−∞} and suppose thatS ⊂Kn is a convex semialgebraic set.
Then, there exists a projected spectrahedron S′ ⊂Kn such thatval(S) =val(S′).
In the second part of the dissertation, we study the relationship between the tropicalizations of convex sets and stochastic mean payoff games. In Chapter 6 we present this class of games and analyze them using Shapley operators [Sha53], Kohlberg’s theorem [Koh80], and the Collatz–
Wielandt property [Nus86]. Our presentation is based on [AGG12]. We give one new result in this area—we generalize the tropical characterization of winning states of deterministic games given in [AGG12] to the case of stochastic games. By combining this result with the results of earlier chapters, we obtain the following correspondence between tropicalization of cones and stochastic mean payoff games.
Theorem E (Theorem 6.30). Let S ⊂ Tn. Then, S is a tropicalization of a closed convex semialgebraic cone if and only if there exists a stochastic mean payoff game such that its Shapley operator F: Tn→ Tn satisfiesS ={x ∈Tn:x ⩽F(x)}. Moreover, the support of S is given by the biggest winning dominion of this game. In particular, S is nontrivial if and only if the game has at least one winning state.
As a consequence of the theorem above, the feasibility problem for any semialgebraic cone over Puiseux series can, in theory, be reduced to solving a mean payoff game. Nevertheless, even though our proofs are constructive, finding the appropriate game is not easy in general.
Indeed, as indicated before, while deciding the winner of a stochastic mean payoff game belongs to the class NP∩coNP, there are problems of unknown complexity that can be expressed as conic semidefinite feasibility problems. However, the tropical polynomial systems mentioned in Theorem C can be converted to Shapley operators. This implies that generic nonarchimedean semidefinite feasibility problems for cones over Puiseux series can be solved by a reduction to mean payoff games. Even more, our approach can solve some semidefinite feasibility problems that do not satisfy the genericity condition. This is proven in Chapter 7 and summarized below.
Theorem F (Theorem 7.6). Suppose thatQ(1), . . . ,Q(n)∈Km×m are symmetric matrices, let Q(x) :=x1Q(1)+· · ·+xnQ(n) andS :={x∈Kn⩾0:Q(x)≽0}. Furthermore, suppose that the matricesQ(k)have rational valuations, val(Q(1)), . . . ,val(Q(k))∈(Q∪{−∞})m×m. Then, given only the signed valuations sval(Q(k)) of these matrices, we can construct (in polynomial-time) a stochastic mean payoff game with the following properties. If the maximal value of the game is strictly positive, then S is nontrivial. If the maximal value is strictly negative, then S is trivial.
Furthermore, if the maximal value is equal to 0 and the matrices val(Q(1)), . . . ,val(Q(n)) are generic, then S is nontrivial. Conversely, solving stochastic mean payoff games can be reduced to the problem of checking the feasibility of spectrahedral cones S ⊂Kn⩾0 as above.
1.2. Our contribution 15 Along the way, we show that the problem of checking the feasibility of a tropical Metzler spectrahedron (defined only by the tropical polynomial inequalities, without the reference to Puiseux series) is equivalent to solving stochastic mean payoff games.
Theorem G (Theorem 7.4). The problem of checking the feasibility of tropical Metzler spec-trahedral cones is polynomial-time equivalent to the problem of solving stochastic mean payoff games.
In the final chapter of the thesis we study the value iteration, which is a simple algorithm that can be used to solve stochastic mean payoff games. This algorithm is based on the fact that if F:Tn → Tn is the Shapley operator of such game, then the limit limN→∞FN(0)/N (where FN = F ◦ · · · ◦F) exists and is equal to the value vector of this game. The value iteration algorithm computes the successive values of FN(0)and deduces the properties of the limit out of this computation. As noted above, the feasibility problem of tropical spectrahedra corresponds to deciding the sign of the value. The following observation provides a condition number for this problem.
Theorem H (Theorem 8.25). Let f: Rn → Rn be monotone and additively homogeneous.
Furthermore, suppose that the equation f(u) =η+u has a solution(η, u)∈R×Rn. Then, we have limN→∞fN(0)/N =η(1,1, . . . ,1). Moreover, suppose that η̸= 0 and letR := inf{∥u∥H∈ Rn:f(u) =η+u}, where ∥ · ∥H is the Hilbert seminorm. Then, for every
N ⩾⌈1 + R
|η|⌉
the entries of fN(0) have the same sign and this sign if the same as the sign of η.
The assumptions of this theorem are fulfilled, for instance, if f is the Shapley operator of a stochastic mean payoff game with constant value. In this case η is the value of the game.
However, there are other class of games for which this result may be of interest, such as the entropy games of [ACD+16, AGGCG17]. When specified to tropical Metzler spectrahedra, the condition numberR/|η|has a geometric interpretation.
Proposition I (Proposition 8.16). Suppose that F:Tn→Tn is a Shapley operator associated with a tropical Metzler spectrahedral cone S ⊂ Tn. Furthermore, suppose that F satisfies the conditions of Theorem H and that η > 0. Then, there exists a tropical Metzler spectrahedral coneS ⊂˜ T2nsuch thatS is the projection ofS˜and such thatη/2is the radius of the largest ball in Hilbert seminorm that is included in S˜. Furthermore, if u∈Rn is such that F(u) =η+u, thenu is the projection of the center of one such ball.
In this way, the quantity |η|measures the width of the tropical Metzler spectrahedral cone, while the quantity R measures the distance of this cone from the origin. Intuitively speak-ing, this should be compared to the quantities that govern the complexity of the ellipsoid method [GLS93]. The complexity of this method depends polynomially on log(R/r), where R is the radius of a ball that contains a given convex body andr is the radius of a ball included in this body. Thus, R measures how far the body is from the origin, while r measures how large it is. The fact that value iteration depends polynomially on R/|η|instead of log(R/|η|) is intu-itively justified by the fact that taking valuation already corresponds to taking the logarithm of the input. The analogy with the ellipsoid is strengthened by the fact that value iteration is also based on an oracle—in order to use this method, it is enough to have an oracle that approximately evaluatesf.
16 Chapter 1. Introduction We also give another application of the condition number, relating the nonarchimedean and archimedean feasibility problems. As discussed earlier, a spectrahedronS ⊂Kn⩾0 over Puiseux series can be seen as a family of real spectrahedraS(t)⊂Rn⩾0 over real numbers. By a general o-minimality argument, there existst0>0such that for allt > t0the feasibility problems forS andS(t)coincide (i.e.,S(t)is nonempty if and only ifSis nonempty). This correspondence may potentially lead to a homotopy-type algorithm for deciding the feasibility of some spectrahedra.
To create such an algorithm, it is desirable to have bounds on the quantityt0. The next theorem shows thatt0 is not big if the associated stochastic mean payoff game is well conditioned.
Theorem J (Theorem 8.4). Suppose that symmetric tropical Metzler matricesQ(1), . . . , Q(n)∈ Tm±×m create a well-formed tropical linear matrix inequality and let η ∈R denote the maximal value o the associated stochastic mean payoff game. LetQ(1), . . . ,Q(n) ∈Km×mbe the monomial lift of Q(1), . . . , Q(n) defied as Q(k)ij :=δijt|Q(k)ij |, where δij := 1 if i=j and δij :=−1 otherwise.
Then, for any
t >(2(m−1)n)1/(2η)
the real spectrahedral cone S(t) := {x ∈ Rn⩾0: x1Q(1)(t) +· · · +xnQ(n)(t)} is nontrivial if and only if the nonarchimedean spectrahedral cone S := {x∈Kn⩾0:x1Q(1)+· · ·+xnQ(n)} is nontrivial.
The main contribution of Chapter 8 is to give explicit bounds for the condition number R/|η|in the case whenf is a Shapley operator of a stochastic mean payoff game. This is based on the following theorem that estimates the bit-size of the invariant measure of a finite Markov chain.
Theorem K (Theorem 8.44). Suppose that P ∈ [0,1]n×n is an irreducible stochastic matrix with rational entries, let π ∈ ]0,1]n be the stationary distribution of P, and let M ∈ N∗ be the common denominator of all the entries of P. Then, the least common denominator of the entries ofπ is not greater that nMmin{nr,n−1}, where nr ⩽n is the number of rows ofP which are not deterministic (i.e., which have an entry in the open interval]0,1[). Moreover, the bound nMmin{nr,n−1} is optimal.
Bounds of similar nature have already appeared in the literature concerning stochastic mean payoff games [BEGM15, AGH18, Con92, ACS14]. However, the proofs in these works are based on the Hadamard inequality that leads to suboptimal results. In order to achieve the optimal bound, we replace the use of the Hadamard inequality by the combinatorial formula of Freidlin and Wentzell [FW12]. As a corollary, we can estimate the pessimistic number of iterations that is needed to solve a stochastic mean payoff game. This gives the following fixed-parameter complexity result for this class of games.
Theorem L (Theorems 8.58 and 8.68). The value iteration algorithm solves constant value games in pseudopolynomial-time and polynomial memory when the number of randomized ac-tions is fixed. Moreover, for general games (with arbitrary value), a modification of the value iteration can find the maximal (or minimal) value of the game, and the set of states that achieve it, in pseudopolynomial-time and polynomial memory when the number of randomized actions is fixed.
We note that the second part of the theorem relies on the characterization of dominions established in Chapter 6. The only other known algorithms that achieve the complexity given in Theorem L is the pumping algorithm studied in [BEGM15] and the variant of the ellipsoid
1.3. Organization of the manuscript 17