Enrique Nadal

Emmanuelle Abisset-Chavanne
email: emmanuelle.abisset-chavanne@ec-nantes.fr

Elias Cueto
email: ecueto@unizar.es

Francisco Chinesta
email: francisco.chinesta@ensam.edu

Model reduction, data-based and advanced discretization in computational mechanics On the physical interpretation of fractional diffusion

Keywords:

Even if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.

Introduction

From standard diffusion to anomalous diffusion

The microscopic nature of diffusion is evident from the pioneering works of Einstein, who considered the increment in the particle position (assumed defined, for the sake of simplicity, in the unbounded one-dimensional axis x) as a random variable, with a probability density given by φ( ). Thus, the particles balance can be expressed by both

ρ(x, t + τ ) = ρ(x, t) + ∂ρ ∂t τ + (τ 2 ) ( 1
)
and

ρ(x, t + τ ) = R ρ(x + , t)φ ( ) d (2)
Developing ρ(x + , t),

ρ(x + , t) = ρ(x, t) + ∂ρ ∂x + 1 2 ∂ 2 ρ ∂x 2 2 + ( 3 )
which, substituted into the right-hand side of Eq. ( 1), and taking into account the normality and expected symmetry

R φ( ) d = 1 R φ( ) d = 0 (4)
leads, after equating Eqs. ( 1) and ( 2), to ρ(x, t) +

∂ρ ∂t τ = ρ(x, t) + 1 2 ∂ 2 ρ ∂x 2 R 2 φ( ) d (5)
or

∂ρ ∂t τ = 1 2 ∂ 2 ρ ∂x 2 R 2 φ( ) d (6)
If we define the diffusion coefficient D as

D = 1 2τ R 2 φ( ) d (7)
the particle balance, also known as the diffusion equation, is given by

∂ρ ∂t = D ∂ 2 ρ ∂x 2 (8)
The integration of this equation, assuming that all the particles are localized at the origin at the initial time, ρ(x,

t = 0) = δ(x), leads to ρ(x, t) = 1 √ 4πDt e -x 2 4Dt (9)
whose second-order moment (variance) scales with time

x 2 = 2Dt (10)
that is, the mean squared displacement scales with the elapsed time t, and the diffusion coefficient D.

The same equation can be derived by describing diffusion as a random walk. Again, for the sake of simplicity, we restrict our discussion to the 1D case, with the x-axis equipped with a grid of size x. In a discrete time step t, the test particle is assumed to jump to one of its nearest neighbor sites, with random direction. Such a process can be modeled by the master equation that writes, at site j, [START_REF] Podlubny | Fractional Differential Equations[END_REF] where W j (t) represents the probability of having the particle at site j at time t and the pre-factor 1/2 accounts for the direction isotropy of the jumps.

W j (t + t) = 1 2 W j+1 (t) + 1 2 W j-1 (t)
By taking classical Taylor expansions,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ W j (t + t) = W j (t) + ∂ W j (t) ∂t t t + ( t 2 ) W j+1 (t) = W j (t) + ∂ W (t) ∂x j x + 1 2 ∂ 2 W (t) ∂x 2 j x 2 + ( x 3 ) W j-1 (t) = W j (t) -∂ W (t) ∂x j x + 1 2 ∂ 2 W (t) ∂x 2 j x 2 -( x 3 ) ( 12
)
that, injected into Eq. ( 11), yield

∂ W ∂t = D ∂ 2 W ∂x 2 (13)
with D defined in the limit of x → 0 and t → 0 by

D = x 2 2 t ( 14
)
which leads to the diffusion equation previously derived.

In complex fluids, micro-rheological experiments often exhibit anomalous sub-diffusion or sticky diffusion, in which the mean square displacement of Brownian tracer particles is found to scale as x 2 ∝ t α , 0 < α < 1 (see [START_REF] Jaishankar | Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations[END_REF] and the references therein). In these cases, the use of non-integer derivatives can constitute an appealing alternative, as it allows one to correctly reproduce the observed physical behavior while keeping the model as simple as possible. Moreover, from a physical point of view, the use of non-integer derivatives introduces a degree of non-locality that seems to be in agreement with the intrinsic nature of the physical system, as discussed later.

As proved in [START_REF] Klafter | Stochastic pathway to anomalous diffusion[END_REF][START_REF] Metzler | The random walks guide to anomalous diffusion: a fractional dynamics approach[END_REF] fractional diffusion models can be derived from continuous time random walks -CTRW -leading to a fractional diffusion equation in the same manner as standard random walks lead to the usual diffusion equation, where subdiffusion is associated with long rest, whereas superdiffusion is related with long jumps. A detailed derivation can be found in [START_REF] Chinesta | A Journey Around the Different Scales Involved in the Description of Matter and Complex Systems[END_REF].

In [START_REF] Aguado | Fractional modelling of functionalized CNT suspensions[END_REF], we reinterpreted, by using fractional diffusion, the rheological findings reported in [START_REF] Ma | The rheology and modelling of chemically treated carbon nanotube suspensions[END_REF], which were difficult to explain using standard models. However, that purely phenomenological route, which assumed non-standard randomizing mechanisms, failed to give a physical picture of the subjacent reality. Thus, despite the fact that a surprising excellent agreement between the predictions and the experimental measurements, both in linear viscoelasticity and relaxation after a step strain, was found, no physical mechanisms were proposed to justify the considered anomalous diffusion.

A physically-based explanation was advanced in [START_REF] Nadal | A physically-based fractional diffusion model for semi-dilute suspensions of rods in a Newtonian fluid[END_REF] where particles involved in the suspension were subjected to the bombardment coming from the neighbor molecules of solvent, a viscous drag and an elastic term related to the interaction with neighboring particles. The stochastic inertia-free simulation proved that the mean square displacement does not scale with the time, but with a power of it, lower than one. The analysis of the covariance reveals that the motion can be assimilated to a fractional Brownian motion, that allowed us to associate such a motion with the solution to a fractional Fokker-Planck diffusion equation.

Other anomalies have been reported and explained through involving fractional models. One of them concerns the fact that the apparent conductivity in almost one-dimensional systems (e.g., carbon nanotubes) or two-dimensional (e.g., graphene) depends on the systems size, being this dependence more intense when the system dimensionality decreases [START_REF] Li | Anomalous heat conduction and anomalous diffusion in one-dimensional systems[END_REF]. These works claimed the need of associating fractional models with fractal media [START_REF]Elasticity of fractal materials using the continuum model with non-integer dimensional space[END_REF][START_REF] Carpinteri | A fractional calculus approach to the mechanics of fractal media[END_REF].

Fractional derivatives

In this section, we propose a general diffusive flux definition based on the use of fractional derivatives. This fractional flux will be coupled with the usual balance equation (which we assume of integer nature). Before making these proposals and developments, we start by revisiting some key results of fractional calculus considered all along the present work.

There exist many good books on fractional calculus and fractional differential equations (e.g., [START_REF] Podlubny | Fractional Differential Equations[END_REF][START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF]). We summarize here the main concepts needed to understand the developments carried out below.

We start with the formula attributed to Cauchy to evaluate the n-th integration, n ∈ N, of a function f (t):

J n f (t) := • • • f (τ ) dτ = 1 (n -1)! t 0 (t -τ ) n-1 f (τ ) dτ (15)
This can be rewritten as [START_REF] Livi | Heat in one dimension[END_REF] where (n) = (n -1)! is the gamma function. The latter being, in fact, defined for any real value α ∈ R, we can define the fractional integral from

J n f (t) = 1 (n) t 0 (t -τ ) n-1 f (τ ) dτ
J α f (t) := 1 (α) t 0 (t -τ ) α-1 f (τ ) dτ (17)
If we consider the fractional derivative of order α and select an integer m ∈ N such that m -1 < α < m, then it suffices to consider an integer m-order derivative combined with a (mα) fractional integral (this is depicted in sketch form in Fig. 1).

Obviously, we could take the derivative of the integral or the integral of the derivative, resulting in the Riemann-Liouville and Caputo definitions of the fractional derivative usually denoted by D α f (t) and D α * f (t), respectively,

D α f (t) = ⎧ ⎪ ⎨ ⎪ ⎩ d m dt m 1 (m-α) t 0 f (τ ) (t-τ ) α+1-m dτ , m -1 < α < m d m f (t) dt m , α = m (18) and D α * f (t) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 (m-α) t 0 d m f (τ ) dτ m (t-τ ) α+1-m dτ , m -1 < α < m d m f (t) dt m , α = m (19)
Because these approaches to the fractional derivative began with an expression for the repeated integration of a function, one could consider a similar approach for the derivative. This was the route considered by Grunwald and Letnikov (GL) defining the so-called "differintegral", which leads to the fractional counterpart of the usual finite differences.

When considering space differential operators, the left-and right-hand-side forms of the Riemann-Liouville and Caputo definitions are used [START_REF] Yang | Numerical methods for fractional partial differential equations with Riesz space fractional derivatives[END_REF]. Thus we consider the definitions of both derivatives

x 0 D α f (x) = d m dx m ⎛ ⎝ 1 (m -α) x 0 f (ξ ) (x -ξ) α+1-m dξ ⎞ ⎠ (20) and L x D α f (x) = d m dx m ⎛ ⎝ 1 (m -α) L x f (ξ ) (ξ -x) α+1-m dξ ⎞ ⎠ (21)
in the case of the Riemann-Liouville, and

x 0 D α * f (x) = 1 (m -α) ⎛ ⎝ x 0 d m f (ξ ) dξ m (x -ξ) α+1-m dξ ⎞ ⎠ (22)
and

L x D α * f (x) = 1 (m -α) ⎛ ⎝ L x d m f (ξ ) dξ m (ξ -x) α+1-m dξ ⎞ ⎠ ( 23
)
when considering the Caputo fractional derivative. One important property that will be used all along this work is the fact that, when the derivative approaches an integer value, the associated expression reduces to its standard integer expression. For the sake of completeness, we reproduce the proof it in the case of the Riemann-Liouville derivative. For that purpose, we consider

t a D α f (t) = d m dt m ⎛ ⎝ 1 (m -α) t a f (τ ) (t -τ ) α+1-m dτ ⎞ ⎠ (24)
that, by integrating by parts, results

t a D α f (t) = d m dt m ⎛ ⎝ (t -a) m-α f (a) (m -α + 1) + 1 (m -α + 1) t a (t -τ ) m-α d f (τ ) dτ dτ ⎞ ⎠ (25)
By taking the limit when α → m leads to

t a D α→m f (t) = d m dt m ( f (a) + ( f (t) -f (a))) = d m f (t) dt m (26)
where, other than the interest of proving that fractional derivatives approach with continuity the ones of integer order, we just proved that, for any function

f (t), when α → m Lim α→m 1 (m -α) t a (t -τ ) m-α-1 f (τ ) dτ = f (t) (27) 1.

Paper outline

The approach based on the use of CTRW remains too microscopic to allow a simple and physical interpretation of the resulting macroscopic fractional equations. In this paper, we focus in superdiffusion and we are following two different routes to extract its physical content, based on the use of the Riemann-Liouville and Caputo fractional derivatives. Next section defines fractional fluxes, the associated fractional diffusion equation, and illustrates the main implications on a simple numerical case. Then, Section 3 proposes a physical view of the Riemann-Liouville -RL -fractional diffusion, while Section 4 addresses a similar analysis on the Caputo fractional diffusion, to conclude with a short discussion on the proposed physical interpretations.

From fractional fluxes to the fractional diffusion equation

The Fourier or Fick equation for standard diffusion writes

Q(x) = -K du(x) dx ( 28
)
where K is the diffusion coefficient.

This equation can be generalized, for non-integer derivatives, in the following way

Q α (x) = - 1 2 K x 0 D α u(x) + L x D α u(x) (29)
and

Q α * (x) = - 1 2 K x 0 D α * u(x) + L x D α * u(x) (30)
for the Riemann-Liouville and Caputo derivatives respectively; it can be noticed that the spatial gradient operator has been symmetrized to represent the symmetric (spatial) physics of diffusion. The fractional diffusion equation results from the mass or energy balances,

∂u ∂t = d dx F(x) (31)
where F(x) refers to the flux at position x ∈ (0, L) ⊂ R, in particular the standard or anomalous fluxes,

Q(x), Q α (x) or Q α * (x).
In what follows, we restrict our analysis to the case where m = 1 and 0 < α ≤ m = 1.

Numerical results

To compare the heat flux behavior in the case of standard and fractional fluxes, we solve the heat equation in the domain (0, L), for L = 1 and L = 5, while enforcing as boundary conditions u(x = 0) = 0 and u(x = L) = L, with conductivity K = 1, α = 1 and α = 0.5 in the standard and fractional flux, respectively.

In the standard case, the solution becomes u(x) = x, and consequently the associated flux u (x) = -1. In the fractional case (when using the Caputo fractional derivative), the expected solution is again u(x) = x, however fluxes are expected to depend on the domain's length. Figs. 2 and 3 corroborate these expectations.

Riemann-Liouville -RL -fractional diffusion

When considering the RL flux Q α , the left contribution to the flux involves the integral

x 0 D α u(x) = d dx ⎛ ⎝ 1 (1 -α) x 0 u(ξ ) (x -ξ) α dξ ⎞ ⎠ (32)
We just proved that when α → 1, it becomes x 0 D α→1 u(x) = du(x) dx , that is, the standard integer derivative, leading to the standard diffusive flux. However, the previous equation reflects the fact that, for any choice of α, real or integer, the flux can be viewed as the result of an integral, even if, in the integer case, the integral collapses to the local gradient (in any case an alternative integral form can be associated with it).

To better apprehend its physical content, we proceed by taking its derivative [START_REF] Molz | A physical interpretation for the fractional derivative in Levy diffusion[END_REF], and even if the derivative will be introduced into the integral, the resulting expression will differ from the Caputo formulation, as proved later.

By applying the Leibniz's rule to the right-hand member of Eq. ( 32), it results (after taking into account that, as proved in [START_REF] Molz | A physical interpretation for the fractional derivative in Levy diffusion[END_REF], the term at x cancels with the right-flux contribution)

x 0 u(ξ ) (x -ξ) α dξ ⎞ ⎠ = - α (1 -α) x 0 1 (x -ξ) α+1 u(ξ ) dξ (33)
where the derivative applies on the kernel without affecting the field u(ξ ), making the difference with respect to the Caputo's expression addressed later.

Thus, the resulting flux at position x, Q α (x), reads

Q α (x) = 1 2 K α (1 -α) ⎧ ⎨ ⎩ x 0 1 (x -ξ) α+1 u(ξ ) dξ - L x 1 (ξ -x) α+1 u(ξ ) dξ ⎫ ⎬ ⎭ (34)
This expression reflects the fact that the field u(ξ ) at position ξ induces a flux at x, independently of the distance between ξ and x, even if, as expected, the weight affecting the contribution of u(ξ ) on the flux at x decreases with the mutual distance, in particular with |xξ | 1+α . This result is not totally surprising, because it seems in agreement with the Einstein rationale exposed in the Introduction or its extended CTRW formulation [START_REF] Chinesta | A Journey Around the Different Scales Involved in the Description of Matter and Complex Systems[END_REF]. In that approach, there is a non-zero probability that a particle located far from x reaches x in a single time step.

If we consider, in Eq. ( 34), both integrals for the limit case α ≈ 1 and assume that the main contribution to both integrals occurs in the interval [x -/2, x + /2], it results

x 0 1 (x -ξ) 2 u(ξ ) dξ - L x 1 (ξ -x) 2 u(ξ ) dξ ≈ x x-/2 1 (x -ξ) 2 u(ξ ) dξ - x+ /2 x 1 (ξ -x) 2 u(ξ ) dξ ≈ 1 ( 2 ) 2 u x - 2 2 - 1 ( 2 ) 2 u x + 2 2 = 2 u x - 2 -u x + 2 (35)
Now, by expressing

u x - 2 = u(x -) + u(x) 2 (36)
and

u x + 2 = u(x + ) + u(x) 2 (37)
introducing both into Eq. ( 35) leads to 2

u x - 2 -u x + 2 = u(x -) -u(x + ) ≈ -2 du(x) dx (38)
which, as expected, is related to the usual gradient with the opposite sign, which combined with Eq. (34), results in the standard flux. The main difference with fractional diffusion is that, when α differs from one, the integral requires richer approximation involving larger intervals around x.

An immediate consequence is that, if the field u(x) is symmetric with respect to x, the net flux vanishes. Another important consequence is that the physics induces a characteristic length, the one characterizing the domain around x, whose contribution to the integral can not be ignored. When α decreases, this characteristic length increases, and then transport properties will depend on the domain size L, a fact which could explain the findings reported in [START_REF] Narayan | Anomalous heat conduction in one-dimensional momentum-conserving systems[END_REF][START_REF] Livi | Heat in one dimension[END_REF]. When α ≈ 0, the integrals involved in Eq. (34) diverge.

To apply now the space derivative involved in the balance equation (31), it suffices to consider the derivative of Eq. (34) using again the Leibniz rule, which implies the second derivative of the kernel function.

Thus, the main consequence of using the Riemann-Liouville derivative is the obtention of a non-local integral form implying that the far field (far with respect to x) affects the flux at position x, in agreement with the microscopic interpretation of diffusion from continuous time random walk, and more particularly its Levy variant. The case of standard diffusion is not an exception; it can be formulated and interpreted within the same framework, even if its final description becomes local.

Caputo fractional diffusion

Now, we perform a similar analysis, but now considering the Caputo fractional derivative. The flux in this case writes

Q α * = - 1 2 K x 0 D α * u(x) + L x D α * u(x) (39) with x 0 D α * u(x) = 1 (1 -α) ⎛ ⎝ x 0 1 (x -ξ) α du(ξ ) dξ dξ ⎞ ⎠ (40)
It is important to note that, by using the previous results, integral (40) results in

x 0 D α→1 * u(x) = du(x) dx ( 41
)
but it is important to highlight its physical integral nature.

Thus, the net flux

Q α * (x) reads Q α * (x) = - 1 2 K 1 (1 -α) ⎧ ⎨ ⎩ x 0 1 (x -ξ) α du(ξ ) dξ dξ + L x 1 (ξ -x) α du(ξ ) dξ dξ ⎫ ⎬ ⎭ (42)
This expression states that the flux at position x depends on the existing field first-order derivative at position ξ with an undoubtable physical content, expressing the flux from the derivatives at different length scales, as discussed later.

The point in which the Caputo route seems advantageous from a conceptual viewpoint concerns its application on the balance equation that, making use of the fractional derivatives composition rule, reads d dx

Q α * (x) = - 1 2 K x 0 D α+1 * u(x) + L x D α+1 * u(x) (43) or d dx Q α * (x) ∝ ⎧ ⎨ ⎩ x 0 1 (x -ξ) α d 2 u(ξ ) dξ 2 dξ + L x 1 (ξ -x) α d 2 u(ξ ) dξ 2 dξ ⎫ ⎬ ⎭ (44)
When α → 1, by using the previous results, Lim α→1 1

(1α)

⎧ ⎨ ⎩ x 0 1 (x -ξ) α d 2 u(ξ ) dξ 2 dξ + L x 1 (ξ -x) α d 2 u(ξ ) dξ 2 dξ ⎫ ⎬ ⎭ = 2 d 2 u(x) dx 2 (45)
which proves that the curvature at x can be from the integral of the weighted curvatures at positions ξ .

Again, when α differs from one, the integral involves larger characteristic integration domains and the final integral does not reduce to a local expression.

If we come back to expression (42), and apply the derivative using Leibniz's rule d dx

Q α * (x) = 1 2 K α (1 -α) ⎧ ⎨ ⎩ x 0 1 (x -ξ) α+1 du(ξ ) dξ dξ - L x 1 (ξ -x) α+1 du(ξ ) dξ dξ ⎫ ⎬ ⎭ (46)
As before, if we consider both integrals in the limit case α ≈ 1 and assume that the main contribution to both integrals occurs in the interval [x -/2, x + /2], it follows

x 0 1 (x -ξ) 2 du(ξ ) dξ dξ - L x 1 (ξ -x) 2 du(ξ ) dξ dξ ≈ x x-/2 1 (x -ξ) 2 du(ξ ) dξ dξ - x+ /2 x 1 (ξ -x) 2 du(ξ ) dξ dξ ≈ 2 du(ξ ) dξ x-2 - du(ξ ) dξ x+ 2 (47)
Now, by expressing both derivatives from the ones existing at x -, x and x + , it results 2 du(ξ )

dξ x-2 - du(ξ ) dξ x+ 2 = 1 du(ξ ) dξ x- - du(ξ ) dξ x+ ≈ -2 d 2 u(x) dx 2 (48)
When α differs from one, the previous discrete approximation is not valid anymore; however, the discrete expression of both integrals in (46) reads d dx

Q α * (x) = 1 2 K α (1 -α) i 1 (x -i ) α+1 du(ξ ) dξ ξ =x-i - i 1 (x + i ) α+1 du(ξ ) dξ ξ =x+i (49) or grouping d dx Q α * (x) = 1 2 K α (1 -α) i β i 1 2i du(ξ ) dξ ξ =x-i - du(ξ ) dξ ξ =x+i ≈ - 1 2 K α (1 -α) i β i C i (x) (50) where C i (x) = du(ξ ) dξ ξ =x+i -du(ξ ) dξ ξ =x-i 2i (51)
which results in a sort of multi-scale curvatures C i all them evaluated at position x and all them contributing to the solution at position x.

Discussion and conclusions

After the previous discussion on the physical content of standard and anomalous diffusion, the last considered under two different perspectives, the ones related to the Riemann-Liouville and Caputo fractional derivatives, the integral nature of diffusion based on these descriptions seems established. When the derivative considered in the flux definition is integer, the integral collapses into a local standard gradient (defining standard flux) or a local standard curvature when applying the divergence of the flux imposed by the balance equation. The remaining question concerns the physics that implies the existence of a non-integer derivative.

A natural explanation consists in assuming that it emerges from larger particle flights than the ones concerned by standard diffusion. For example, when considering Levy flights, the jump length variance diverges.

By using the Caputo formulation, which considers the flux at position x from the contribution of a sequence of gradients involving different characteristic lengths (discrete or continuously distributed), one could expect that, in the presence of multi-scale or fractal media, the flux at position x is composed by the contribution of the different-scale fluxes, justifying the integral expression.

In the case when considering almost one-dimensional media, particles at position x exhibit a spectrum of correlated behavior, and consequently again the gradients involved in the fluxes should concern different characteristic lengths, explaining once more the integral expression of the fluxes. When dimensionality increases, long-distance correlations disappear, because of the interactions coming from the new dimension(s) and consequently physics regains its locality.

Fig. 1 .

 1 Fig. 1. Schematic cartoon of the derivative of order 1.7.

Fig. 2 .

 2 Fig. 2. Standard flux model with K = 1: (top) temperature field; (bottom) flux field; (left) L = 1 and (right) L = 5.

Fig. 3 . 1 ( 1

 311 Fig. 3. Fractional flux model with K = 1 and α = 0.5: (top) temperature field; (bottom) flux field; (left) L = 1 and (right) L = 5.


Emilie Courvoisier

published or not. The documents may come

Multi-scale and multi-technique analysis of the thermal degradation of poly(ether ether ketone)


Goulwen Fichou
email: goulwen.fichou@univ-rennes1.fr

Jean-Philippe Monnier
email: jean-philippe.monnier@univ-angers.fr

Ronan Quarez
email: ronan.quarez@univ-rennes1.fr

Ronan Quarez

SUBSTITUTION PROPERTY FOR THE RING OF CONTINUOUS RATIONAL FUNCTIONS

Keywords: continuous rational function, real algebraic variety, substitution property

We study the substitution property for the ring R 0 (V ) of continuous rational functions on a real algebraic affine variety V . We show that R 0 (V ) satisfies a substitution property along points; moreover, when V is non-singular, it satisfies also a substitution property along Puiseux arcs, which characterizes R 0 (V ).

Introduction

Given a real algebraic variety V ⊂ R n , any morphism φ : R[V ] → R can be seen as an evaluation morphism at a certain point x 0 ∈ V , where R[V ] denotes the coordinate ring of V . For any extension ring B of R[V ] a natural question is to ask whether the evaluation morphism does extend uniquely to B. This question is known as the substitution property, when one considers, more generally, an evaluation morphism φ : R[V ] → R into a real closed field extension R of R. Such a property gives a lot of information on the real algebra of the ring B namely, one may derive Artin-Lang property, Positivstellensatz, etc.

A natural class of rings B to test are the different rings of functions considered in real algebraic geometry. The larger class consists in the ring of continuous (with respect to the Euclidean topology) semi-algebraic functions on V , where the continuity is intended with respect to the Euclidean topology on V . In that situation, the substitution property is known to be true by [START_REF] Schwartz | Real closed rings, Habilitationsschrift[END_REF]. A more rigid class is given by the ring of Nash functions on V , namely real analytic and semi-algebraic functions. Here also the substitution property holds true (see [2, 8.5.2]).

In the present paper, we study the case of the ring R 0 (V ) of continuous rational functions on V . Continuous rational functions on a real algebraic variety V are those rational functions that admit a continuous extension along their poles. They form an intermediate ring between regular functions (rational functions without real poles) and continuous semi-algebraic functions. This ring has been intensively studied in several recent works ( [START_REF] Fichou | Continuous functions on the plane regular after one blowing-up[END_REF], [START_REF] Kollár | Continuous rational functions on real and p-adic varieties[END_REF], [6], [START_REF] Kucharz | Rational maps in real algebraic geometry[END_REF], [START_REF] Kucharz | Stratified-algebraic vector bundles[END_REF], [START_REF] Kollar | Curve-rational functions[END_REF], [START_REF] Fichou | Fonctions régulues[END_REF], [START_REF] Monnier | Semi-algebraic geometry with rational continuous functions[END_REF]).

The main results of the paper are the following. We first show that R 0 (V ) satisfies the substitution property (Theorem 2.13), generalizing a result in [START_REF] Fichou | Continuous functions on the plane regular after one blowing-up[END_REF]. A natural question is then whether this property characterizes the ring of continuous rational functions. However we prove that this is not the case (Proposition 2.14). This motivates the introduction of the concept of evaluation along Puiseux arcs in section 3. More precisely, we take evaluation morphisms of the form φ : R[V ] → R[[t 1/ N ]], where R[[t 1/ N ]] alg is the ring of Puiseux power series over R that are algebraic over real polynomials. Depending upon our variety V is singular or not, we obtain two opposite results. Firstly, if V is non-singular, then R 0 (V ) satisfies the substitution property along arcs and moreover this property characterizes R 0 (V ) (see theorem 3.5). Secondly, it exists a singular variety V such that R 0 (V ) does not satisfy the substitution property along arcs (see theorem 4.7). Definition 2.1. Let A be an R-algebra and B be an A-algebra. We say that B has the substitution property over A if, for any real closed field R, any morphism A → R admits one and only one lifting to B, namely such that the following diagram is commutative :

Substitution on points

A → R ↓ ր B
If the property if valid for the particular real closed field R = R we say that B satisfies the weak substitution property.

Recall that an ideal I of A is called real if, for every sequence a 1 , . . . , a k of elements of A, then a 2 1

+ • • • + a 2 k ∈ I implies a i ∈ I for i = 1, . . . , k. A field F is called real if (0) is a real ideal in F .
Here is the first easy fact :

Proposition 2.2. Let us assume that A is a domain whose fraction field is real. If B satisfy the substitution property over A, then the morphism A → B is injective.

Proof. Let us assume that 0 = a ∈ A is sent onto 0 in B. Since the fraction field of A is real, the null ideal (0) in A is real and hence it is the intersection of all the real prime ideal of A.

Then, there is a real prime ideal p in A such that a / ∈ p. Take then for R the real closure of the residual field at p. Then, a is sent onto a non-zero element in R, which leads to a contradiction since it should be 0 after lifting the evaluation morphism to B.

In the following, we will mainly consider rings B which are subrings of the fraction field of a domain A. If f = p/q where p and q lye in A, starting from a morphism φ : A → R, it is easy to define φ(f ) = φ(p) φ(q) as long as φ(q) = 0. This elementary observation gives us the substitution property if B is a subring of the ring of regular functions of A. We recall that the ring of regular functions of A is just the ring

(1 + A 2 ) -1 A.
Hence, in all the following, we will mainly consider rings B which are subrings of the field of rational functions and which contain the ring of regular functions of A. And the problem we will face is to define φ(f ) when φ(q) = 0.

The first important consequence of definition 2.1, is that one gets a one-to-one correspondence between the real spectrum of B and A. For background about the real spectrum of a ring, we refer to [START_REF] Bochnak | Real algebraic geometry[END_REF]. In few words, let us say that a point α of the real spectrum of the ring A is an equivalence class of morphisms π from A into a real closed field for the equivalence relation generated by the following : π ∼ π ′ where π : A → R and π ′ : A → R ′ with R and R ′ are real closed field, if there exists a factorization of π ′ through a morphism R → R ′ .

The real spectrum is endowed with two natural topologies : the real spectrum topology and the constructible topology which are defined as follows. Any element a ∈ A can be evaluated at any α ∈ Spec r A simply by evaluating it at the morphism π : A → R given by α. Then, {α ∈ Spec r A | a(α) > 0} gives a basis of open neighborhood for the real spectrum topology. Then, the constructible open subsets are those obtained by boolean combination from the open subsets with respect to the real spectrum topology.

To any semi-algebraic subset S of R n is canonically associated a constructible subset S of Spec r R[x 1 , . . . , x n ]. This association is compatible with the boolean operations (intersection, union, complementary) and is such that the semi-algebraic subset

S = {x ∈ R n | p(x) > 0} where p ∈ R[x 1 , . . . , x n ] is associated to S = {α ∈ Spec r R[x 1 , . . . , x n ] | p(α) > 0}.
Now we state the announced correspondence.

Proposition 2.3. If B satisfy the substitution property over A, then the induced morphism on the real spectrum Spec r B → Spec r A is bijective and continuous (with respect to both the real spectrum topology and the constructible topology).

Proof. By definition, the induced morphism on the real spectrum Spec r B → Spec r A is continuous with respect to the real spectrum topology and also for the constructible topology (see [2, 7.17]). Surjectivity comes immediately from the existence of the factorization in the substitution property.

Injectivity comes immediately from the uniqueness of the factorization in the substitution property. Indeed, let us consider two points β : B → R and β ′ : B → R ′ where R, R ′ are real closed fields and assume that the compositions β • φ and β ′ • φ by φ : A → B give rise to the same point in Spec r A. Namely, there exists a real closed filed R ′′ which extends both R and R ′ , which implies that β = β ′ .

From now on, we will takes for A a ring of polynomials functions (or coordinate ring) R[V ], or one of its localization, where V is an algebraic variety in R n in the sense of [START_REF] Bochnak | Real algebraic geometry[END_REF].

There are two well-known classes of functions whose ring satisfies the substitution property : Nash functions and continuous semi-algebraic functions.

More precisely, if A = R[X 1 , . . . , X n ], then one has the substitution property for the ring of Nash functions on R n ([2, 8.5.2]) and one may generalize it to any smooth algebraic variety V in R n .

The substitution property is also valid in the ring of semi-algebraic continuous functions on V . This result can be seen as a consequence of the theory of real closed rings developed by N. Schwartz in [START_REF] Schwartz | Real closed rings, Habilitationsschrift[END_REF].

See also the work of J. Fernando on the subject [START_REF] Fernando | On the sustitution theorem for rings of semi-algebraic functions[END_REF]. In these two cases, since a morphism A → R corresponds to the evaluation at some point (x 1 , . . . , x n ) in V ⊂ R n , the substitution property says that any evaluation morphism can be uniquely lifted to the ring B.

Beware that a subring of a ring satisfying the substitution property does not necessarily satisfies the substitution property as it is shown by the following example :

Example 2.4. Let A = R[x] and B = R[x, √ 1 + x 2 ] ≃ R[x, y]/(y 2 -(1 + x 2 )).
Then, the morphism A → R which send x to 0 (the evaluation morphism at the origin) admits two liftings to B : the morphism which send y to 1 and the morphism which send y to -1. Moreover, B is a subring of the ring of semi-algebraic continuous functions on R.

2.2.

Substitution for continuous rational functions. Let V ⊂ R n be a real algebraic variety. We will mainly be interested in the ring of continuous rational functions defined on V . These functions are rational functions on V which can be seen as functions defined on the complementary of the zero set of their denominators, and which admits a continuous extension to the whole V . In order to avoid pathological cases, we will mainly assume that the variety V is central which means that the set of non-singular points of V is dense in V for the euclidean topology. Of course, any non-singular variety is central.

Since we require for our rings to be contained in the field of fraction R(V ) of V , one should consider the natural assumption for the variety V to be moreover irreducible. So let us give now the formal definition of continuous rational functions. Definition 2.5. Let V ⊂ R n be an irreducible central real algebraic variety. A continuous function f : V → R is said to be continuous rational if there exists a non-empty Zariski open subset W ⊂ V such that the restriction f |W of f to W is a regular function on W . We denote by R 0 (V ) the ring of continuous rational functions on V .

Remark 2.6. An alternative ring of functions to work with would be the ring of hereditarily continuous rational or regulous functions on a central variety V defined in [START_REF] Kollár | Continuous rational functions on real and p-adic varieties[END_REF] as those continuous rational functions which remains rational in restriction to any subvariety. When the variety V is smooth, we recover continuous rational functions, but in general it leads to a proper subring. We will consider this ring solely in Theorem 2.13.

Let A = R[X 1 , . . . , X n ].
Let us start with an elementary observation about rings of functions satisfying the substitution property. If f is a real function defined on a subset of R n , we denote by Z(f ) the zero set of f . Proposition 2.7. Let A = R[x 1 , . . . , x n ] and B be a sub A-algebra of R(x 1 , . . . , x n ) satisfying the substitution property over A.

Let f = p/q ∈ B with p and q coprimes. Then, (i) Z(q) ⊂ Z(p), (ii) Z(q) has codimension at least 2 in R n . Proof. To show (i), let us just write q = f p and use the substitution property for B over A.

To show (ii), assume there exists an irreducible factor q 1 of q corresponding to a codimension one variety in R n . Then, the condition Z(q 1 ) ⊂ Z(p) implies that q 1 divides p since (q 1 ) is a real prime ideal (cf the so-called change of signs criterion [2, 4.5.1]), in contradiction with the coprimality of p and q.

Remark 2.8. Note that in the one-dimensional case, condition (i) says already that B is a subring of the ring of regular functions.

Note that the previous proposition generalizes to non-singular varieties :

Proposition 2.9. Let A = R[V ]
where V is a non-singular irreducible algebraic variety in R n . Let B be a sub A-algebra of R(V ) satisfying the substitution property over A.

Let f = p/q ∈ B with p and q coprimes i.e (p) + (q) = 1. Then, (i) Z(q) ⊂ Z(p), (ii) Z(q) has codimension at least 2 in V .

Proof. The coordinates ring R[V ] mx of the variety V localized at any point x ∈ V is regular and hence an UFD, so that the proof can be carried out similarly.

This result can be used when B = R 0 (V ) where V ⊂ R n is non-singular (see theorem 2.12). On the contrary, the proposition cannot be extended to the case V is singular as illustrated by the example of the Cartan umbrella.

Example 2.10. Let V ⊂ R 3 with equation x 3 = z(x 2 + y 2 ), and consider the rational function on V given by f = x 3 /(x 2 + y 2 ) extended by z on the stick of the umbrella. Then, the zero set of its denominator q is the whole stick of the umbrella, therefore of codimension one in V .

We come now to the central result of this section, which gives the substitution property for the ring of continuous rational functions on a non-singular real algebraic variety. We recall that the argument given in [4, 5.4] relies on Łojaciewicz inequality together with the fact that a continuous rational function on R n admits a constructible stratification such that it is regular in restriction to its strata.

This last property extend to any non-singular variety:

Theorem 2.11. Let V ⊂ R n be a non-singular irreducible real algebraic variety and f be a continuous rational function on V . Then, there exists a stratification of V into Zariski locally closed subsets S 1 , . . . , S m such that the restriction f | S k of f to S k is a regular function.

Proof. The proof is heavily based on the fact that, since V is non-singular, f is hereditarily rational (regulous) and hence one may use [10, Théorème 4.1].

Let us show now the desired substitution property for continuous rational functions:

Theorem 2.12. Let V ⊂ R n be a non-singular irreducible real algebraic variety. Then, R 0 (V ) satisfies the substitution property over R

[V ]. Moreover, the induced morphism Spec r R 0 (V ) → Spec r R[V ]
is an homeomorphism with respect to the constructible topology.

Proof. We focus on the unicity of the factorization, since the existence comes from the usual evaluation. First, let us see what happens if R = R. Then, one may assume, for simplicity, that the origin o is in V and that φ(x 1 ) = . . .

= φ(x n ) = 0. Let f ∈ R 0 (V ). Up to considering f -f (o), one may assume that f (o) = 0.
Then, by Łojasiewicz property [2, Proposition 2.6.4] applied to x 2 1 +. . .+x 2 n and f , there exists an integer N and a continuous rational function g in R 0 (V ) such that f N = (x 2 1 + . . . + x 2 n )g. This algebraic identity implies that φ(f ) = 0 which concludes the proof for the case R = R.

Let us now consider a general real closed field R and denote by α the point of the real spectrum of R[V ] given by the morphism φ. According to theorem 2.11, we know that, for any f ∈ R 0 (V ), there is a Zariski locally closed stratification V = S 0 ∪ . . . ∪ S m such that f is regular on each S i . Let us assume that α ∈ S 0 ; if we set

S 0 = {x ∈ V | r(x) = 0, s(x) = 0}
where r, s are polynomials, then α ∈ S 0 just means that φ(r) = 0 and φ(s) = 0.

Then, the zero set of r is contained in the zero set of s • (qf -p) where p, q are polynomials such that f = p/q is regular on S 0 . Again, by Łojasiewicz property one gets the existence of an integer N and a continuous rational function g such that (s(qf -p)) N = rg. Applying φ, and since φ(s) = 0 and φ(q) = 0 (since p/q is regular on S 0 ), one gets φ(f ) = φ(p)/φ(q) which concludes the first part of the proof.

By proposition 2.3, the map ϕ : Spec r R 0 (V ) → Spec r R[V ] is continuous and bijective. Let us consider a constructible set T in Spec r R 0 (V ) which is a boolean combination of sets of the form S = {f = 0, g > 0} where f, g ∈ R 0 (V ). Again by theorem 2.11, one knows that there is a Zariski locally closed stratification V = S 0 ∪ . . . ∪ S m such that f and g are regular on each S k . Namely, one may write on S k , f = p k /q k and g = r k /s k where p k , q k , r r , s k are polynomials such that q k and s k do not vanish on S k .

Hence, the image of S by ϕ is then the union of all S k ∩ ϕ(S) which can be written as the subset

S k ∩ {p k = 0, r k s k > 0} in Spec r R[V ]. Finally, one gets that ϕ(T ) is a constructible subset in Spec r R[V ].
The previous proposition says that if a R-algebra homomorphism from R 0 (V ) is the evaluation at a given point in restriction to the polynomials, then it is still the evaluation at this point on any continuous rational functions on V .

So far it does not seem very clear how to get the substitution property for the ring of continuous rational functions on a singular variety since one no longer dispose of the decomposition of rational functions as regular functions on strata.

Although, one may obtain formally, by taking some quotients in a commutative diagram, that the ring of regulous functions on a singular variety V satisfies the substitution property over R [V ]. By definition, the ring of regulous functions on real algebraic variety V ⊂ R n , given by an ideal

I ⊂ R[X 1 , . . . , X n ], is just the quotient ring R 0 (V ) = R 0 (R n )/ I R 0 (R n ) (V ) where I R 0 (R n ) (V ) = {f ∈ R 0 (R n )| V ⊂ Z(f )}.
Hence, one may easily derive the substitution property for the ring of regulous functions on V . Namely:

Theorem 2.13. Let V ⊂ R n be a central irreducible real algebraic variety whose coordinate ring is R[V ] = R[X 1 , . . . , X n ]/I. Then R 0 (V ) satisfies the substitution property over R[V ]. Proof. Let A = R[X 1 , . . . , X n ], B = R 0 (R n ) and I = I R 0 (R n ) (V ).
The following commutative diagram is a consequence of the universal property for quotient rings :

A → A/I → R ↓ ↓ ր B → B/ I ↓ R(V )
Namely, given an evaluation morphism A/I → R it gives rise to a unique morphism A → R which sends I onto 0. Hence, by the substitution property for B, one has an unique lifting morphism B → R which send I onto 0. Hence, by the universal property for quotient rings, one has a unique factorization B/ I → R.

The substitution property does not characterize continuous rational functions.

We show that the ring of continuous rational functions is not the biggest subring of rational functions that satisfies the substitution property and hence, the substitution property is not a characterization for the ring of continuous rational functions. This result will lead to the notion of substitution along arcs in the next section for which such a characterization will be available.

But for the classical substitution property, already in the case of the plane one has:

Proposition 2.14. The ring R 0 (R 2 ) is not maximal in R(x, y) to satisfy the substitution property.

Before entering into the details of the proof, we will state as an intermediary step that there exist bigger rings that satisfy only the existence condition of the lifting in the substitution property. Namely, Proposition 2.15. The ring R 0 (R 2 ) is not maximal in R(x, y) among rings satisfying the existence condition of the lifting in the substitution property. Moreover, there does not exist a unique maximal ring satisfying the existence condition of the lifting in the substitution property.

Proof. For the first point, let us show that the ring R 0 (R 2 )[ x x 2 +y 2 ] satisfies also the existence condition of the lifting in the substitution property.

It suffices to show that the evaluation morphism at the origin φ : R 0

(R 2 ) → R can be lifted to R 0 (R 2 )[ x x 2 +y 2 ]
. The first step in the proof is to show that one may set φ

x x 2 +y 2 = 0. Consider the morphism R 0 (R 2 )[T ] ψ -→ R(x, y) defined by T → x x 2 +y 2 . One has R 0 (R 2 )[ x x 2 + y 2 ] ≃ R 0 (R 2 )[T ]/Ker ψ.
Let P ∈ Ker ψ, and write P = a 0 +. . .+a n T n where the a i 's are continuous rational functions on R 2 . In R(x, y)[T ], one may factorize P by T -p/q where p = x, q = x 2 + y 2 . Hence

(qT -p)(b 0 + . . . + b n-1 T n-1 ) d = a 0 + . . . + a n T n
where the b i 's and d are polynomials in R[x, y].

Then,

( * )            -b 0 p = a 0 d qb 0 -b 1 p = a 1 d . . . qb n-2 -b n-1 p = a n-1 d qb n-1 = a n d
Let r be the valuation of d with respect to the prime p = x, namely v p (d) = r. Since p and q are coprime, one gets from the last identity in ( * ) that v p (b n-1 ) ≥ r since (p) is a real prime ideal. Likewise, v p (b n-2 ) ≥ r and more generally v p (b i ) ≥ r for i = 0, . . . , n -1. The first identity in ( * ) implies that we may write a 0 = h/k with h, k ∈ R[x, y], v p (a 0 ) ≥ 1 and v p (k) = 0. It follows that a 0 vanishes on a non-empty Zariski open subset of Z(x) and thus on whole Z(x). Hence a 0 (0) = 0.

Hence, the setting φ

x x 2 +y 2 = 0 is compatible with any algebraic relation verified by f = p q = x x 2 +y 2 over R 0 (R 2 ). It defines then a lifting of φ to R 0 (R 2 )[ x x 2 +y 2 ]. Note that, if one set φ α x x 2 +y 2 = α ∈ R, one may replace f with g = f -α, namely replace
x with p = x -α(x 2 + y 2 ) which remains real prime. One can then repeat the previous process to get that a 0 (0) = 0 for any algebraic relation a 0 + . . .+ a n g n = 0 where the a i 's are continuous rational functions. Since φ α (g) = 0, again one has a lifting of

φ α to R 0 (R 2 )[ x x 2 +y 2 ].
Hence, one has infinitely many liftings of φ to R 0 (R 2 )[ x x 2 +y 2 ]. Let us show now the second point. Set f = x x 2 +y 2 and g = y x 2 +y 2 . Let B 1 be a ring that contains R 0 (R 2 )[f ] and that satisfies the existence in the substitution property. Then g cannot belong to B 1 . Indeed, from the identity xf + yg = 1 we get, using that φ is the evaluation at the origin, that

1 = φ(xf + yg) = φ(x)φ(f ) + φ(y)φ(g) = 0.
Likewise, let B 2 be a ring that contains R 0 (R 2 )[g] and that satisfies the existence in the substitution property. As previously, one has f / ∈ B 2 . Then, there does not exist a unique maximal ring satisfying the existence condition of the lifting in the substitution property.

The proof of Proposition 2.14 is obtained by slightly modifying the above proof.

Proof of Proposition 2.14. Let us consider the ring

B = R 0 (R 2 )[ x (x 2 + y 2 ) n , n ∈ N * ]. Setting f n = x (x 2 +y 2 ) n , one has f n = f n+1 (x 2 + y 2 ).
Let us note that it suffices to consider the evaluation morphism at the origin A = R[x, y] φ → R, and show that it admits one and only one lifting to B.

If such a lifting (again denoted by φ) of the evaluation at the origin exists, it satisfies necessarily φ(f n ) = φ(f n+1 )φ((x 2 + y 2 )). Since φ(x 2 + y 2 ) = 0, it follows that φ(f n ) = 0 for any n. This shows the unicity of the lifting.

To show the existence, let us consider an algebraic relation of the form P (f 1 , . . . f n ) = 0 where P = a 0 + α>0 a α x α 1 . . . x αn is a polynomial whose coefficients are continuous rational functions on R 2 . The relation P (f 1 , . . . , f n ) = 0 can be rewritten as

a 0 + α>0 a α x α 1 +...+αn (x 2 + y 2 ) α 1 +2α 2 +...+nαn = 0, which implies a 0 (x 2 + y 2 ) N = - α>0 a α (x 2 + y 2 ) N (α) x α 1 +...+αn
where N and N(α) are integers.

From the previous identity, it follows that a 0 vanishes on Z(x) and one gets φ(a 0 ) = 0. Hence φ(P (f 1 , . . . , f n )) = 0 as desired.

In summary, the substitution property as defined in 2.1 is not strong enough to characterize continuous rational functions. More precisely, the lifting property is not sufficient to implies continuity, even if the evaluation morphism can be taken over a Puiseux series field. For instance, we may consider a rational function f = p/q on the plane such that p(0) = q(0) = f (0) = 0, and an algebraic plane curve at the origin parametrized by two Puiseux series γ = (α(t), β(t)) which can also be seen as an element of the real spectrum of R[x, y]. If γ goes to the origin by assumption, it seems not obvious to deduce that f (γ) also goes to the origin.

These considerations lead us to consider a new substitution property along (convergent) arcs in the remaining of the paper.

Substitution along arcs

The ring of all formal power series over R in the indeterminate t will be denoted by R

[[t]].

The ring of all formal power series over R in the indeterminate t which are algebraic over R[t] will be denoted by R[[t]] alg .

We consider moreover several power series rings which are all subrings of the field of Puiseux power series. The field of all Puiseux power series over R in the indeterminate t will be denoted by R((t 1/ N )), and its valuation ring by R[[t 1/ N ]]. The subfield of Puiseux series which are algebraic over R[t] will de denoted by R((t 1/ N )) alg , and its valuation ring by R[[t 1/ N ]] alg . From a geometric point of view, an element in the field R((t 1/ N )) alg is identified with a continuous semi-algebraic functions germ (0, +∞) → R on the right at the origin, the ring R[[t 1/ N ]] alg consisting of those germs which can be extended continuously at the origin [START_REF] Bochnak | Real algebraic geometry[END_REF][Section 8.1].

We will use frequently in the sequel the following identification between arcs traced on a variety V , and R-algebra morphisms R[V ] → R[[t]] alg . More precisely, if V ⊂ R n is an algebraic variety defined by the polynomial equations f 1 = . . . = f r = 0, an algebraic formal arc γ(t) = (γ 1 (t), . . . , γ n (t)) ∈ (R[[t]] alg ) n traced on the variety V satisfies f 1 (γ(t)) = . . . = f r (γ(t)) = 0. In other words, an algebraic formal arc on V is given by an R-algebra morphism R

[V ] → R[[t]] alg .
3.1. Definition and first properties. We state a substitution property along algebraic formal arcs and algebraic Puiseux arcs. Definition 3.1. Let A be an R-algebra and B be an A-algebra. We say that B has the substitution property over A along algebraic formal arcs (respectively algebraic Puiseux arcs) if any morphism

A → R[[t]] alg (resp. R[[t 1/ N ]
] alg ) admits one and only one lifting to B.

Again, we mainly consider the case where A is a ring of polynomials functions. One has the substitution property along algebraic Puiseux arcs for the ring of Nash functions and the ring of continuous semi-algebraic functions. Proposition 3.2. Let A = R[V ] be the coordinate ring of an non-singular real algebraic variety V ⊂ R n . Let B be either the ring of semi-algebraic functions on V or the ring of Nash functions on V . Then B satisfies the substitution property along Puiseux arcs.

Proof. First of all, note that since all the possible rings B being subrings of the ring of semialgebraic continuous functions on V , the evaluation along an arc given by A → R[[t 1/ N ]] alg can always be lifted to B by considering the usual composition of semi-algebraic arcs.

To prove the uniqueness, recall that R[[t 1/ N ]] alg is included into the field of algebraic Puiseux series R((t 1/ N )) which is a real closed field. Then, let us use the fact that the ring B satisfies the substitution property on points (as it has been recalled in subsection 2.1). Since the composite morphism A → R[[t 1/ N ]] alg → R((t 1/ N )) alg admits a unique lifting to B, so it is the case for a given morphism

A → R[[t 1/ N ]] alg .
In the sequel, we will mainly consider rings B which are subrings of the field of rational functions of an irreducible variety, and also rings containing the ring of regular functions (since those later obviously satisfies the substitution property over arcs). Proposition 3.3. If B satisfies the substitution property along algebraic formal or Puiseux arcs over A, then B satisfies also the weak substitution property on points over A.

Proof. It suffices to compose with the evaluation morphism R[[t]

] alg → R (respectively with R[[t 1/ N ]] alg → R) of an arc at its origin, by sending t onto 0.

Let us also mention an example where the weak substitution on points does not imply the substitution along arcs.

Example 3.4. Let A = (R[x]) (x) and B = (R[x, y]/(x 2 -y 2 )) (x,y) .
Then, B has the weak substitution property along points over A but does not satisfy the substitution property over A along arcs. Indeed, the arc x = t written on A can be lifted to two different arcs γ 1 = (t, t) and γ 2 = (t, -t) on B. Moreover, there is only one morphism A → R : the one which sends x to 0. This morphism admits one and only one lifting to B, namely the morphism given by x → 0 and y → 0.

3.2.

Substitution along arcs for continuous rational functions. We show that the ring of continuous rational functions on a non-singular algebraic variety satisfies the substitution property along arcs. Moreover, we show that it is the biggest ring contained in the field of rational functions that one can expect to satisfy this property.

In other words, the substitution property along arcs characterizes, on non-singular varieties, continuous rational functions. It enlightens the importance of substitution along arcs for hereditarily rational functions, as it has already be pointed out in the work of [START_REF] Kollar | Curve-rational functions[END_REF] for instance. Theorem 3.5. Let A = R[V ] be the coordinate ring of a non-singular irreducible real algebraic variety V ⊂ R n . Then the ring R 0 (V ) of continuous rational functions on V satisfies the substitution property along arcs. Moreover, if B is a subring of the field R(V ) of rational functions on V which satisfies the substitution property along arcs, then B ⊂ R 0 (V ).

Proof. First of all, the natural lifting given by the evaluation of a continuous semi-algebraic function along a semi-algebraic arc gives the existence property. For the unicity, remark that R 0 (R n ) satisfies the substitution property on points (theorem 2.13). Then, any morphism

γ : A → R, where R = R[[t 1/ N ]] alg or R[[t]
] alg , gives rise to a morphism A → R((t 1/ N )) alg which can uniquely be lifted to R 0 (R n ). This shows the unicity for our desired substitution property, namely there cannot exist another lifting than the natural one.

Let us consider now a ring B satisfying the substitution property along arcs and let us show the inclusion B ⊂ R 0 (V ).

Let us first consider the case R = R[[t 1/ N ]] alg . One may use the classical fact (see for instance [13, Chap 6, Lemma 4.2]) that a semi-algebraic function f is continuous on S if and only if f • γ is continuous for all algebraic Puiseux arc γ supported in S.

In fact, it is possible to adapt a little bit this result for our purpose :

Lemma 3.6. Let f : S → R be a semi-algebraic function where S is a semi-algebraic subset which has dimension n at the point x 0 ∈ S. Assume that x 0 ∈ T where T is a semi-algebraic subset of dimension < n. Then, f is continuous at x 0 if and only if f • γ is continuous at 0 for any algebraic Puiseux arc γ supported in S \ T ∪ {x 0 } and passing through x 0 .

Proof. If f is not continuous at x 0 , then there is ǫ > 0 such that the set

{ x -x 0 | x ∈ S, |f (x) -f (x 0 )| ≥ ǫ}
contains arbitrarily positive elements. Up to re-sizing this ǫ, using the density of S \ T in T , one gets that

{ x -x 0 | x ∈ S \ T ∪ {x 0 }, |f (x) -f (x 0 )
| ≥ ǫ} contains also arbitrarily positive elements. Since this set is semi-algebraic, it contains an interval I with endpoints 0. Now, by the curve selection lemma, there is an algebraic Puiseux arc γ : I → S \ T ∪ {x 0 } such that γ(t) -x 0 < ǫ and |f (γ(t)) -f (x 0 )| ≥ ǫ. This concludes the proof. Let f = p/q in B and let γ be an algebraic Puiseux arc well-defined at the origin, which is not contained into the polar locus of f , namely such that q • γ = 0. Then, f • γ = p • γ/q • γ is well defined in R((t 1/ N )) alg , and necessarily p • γ/q • γ has a non-negative valuation. Hence, f • γ is continuous.

Moreover, using the substitution property at the origin of the arc, one deduces that the limit is f (γ(0)). In conclusion, f ∈ R 0 (V ) and hence B ⊂ R 0 (V ).

Let us consider now the case R = R[[t]] alg . Let f ∈ B and as previously γ be an algebraic Puiseux arc, well-defined at the origin, which is not contained in the polar locus of f . There exist an integer m such that δ(t) = γ(t m ) ∈ (R[[t]] alg ) n . Then, since B satisfies the substitution property along arcs for R = R[[t]] alg , the arc δ uniquely lifts to B and f • δ is continuous and admits f (δ(0)) as a limit. Composing with the continuous semi-algebraic function t → m √ t on one gets that f • γ is continuous and admits f (γ(0)) as a limit too.

One natural question is what happens if we replace algebraic Puiseux arcs with formal Puiseux series. It happens that the substitution property remains true for continuous rational functions.

Theorem 3.7. Let A = R[V ] be the coordinate ring of a non-singular irreducible real algebraic variety V ⊂ R n . Then the ring R 0 (V ) of continuous rational functions on V satisfies the substitution property along arcs in

R[[t 1/ N ]].
Moreover, if B is a subring of the field R(V ) of rational functions on V which satisfies the substitution property along arcs in

R[[t 1/ N ]], then B ⊂ R 0 (V ). Proof. Let us consider γ : R[V ] → R[[t 1/ N ]]. One may see γ as a morphism into R((t 1/ N ))
which is a real closed field. Then, since R 0 (V ) satisfies the substitution property over R[V ] (theorem 2.12), one gets the unicity for a lifting of γ to R 0 (V ).

In order to prove the existence of such a lifting, we show that the unique lifting of γ with values in R((

t 1/ N )) has, in fact, values in R[[t 1/ N ]]
. By the contrary, let us assume that there is a continuous rational function

f such that γ(f ) / ∈ R[[t 1/ N ]]
. By theorem 2.11, there is a Zariski locally closed stratification V = S 1 ∪ . . . ∪ S t such that f is regular with restriction to each stratum. One derives a constructible stratification of the real spectrum of R[V ], namely V = S 1 ∪ . . . ∪ S t . Moreover, the morphism γ defines a point of V that we assume to lye in S 1 . Since f is regular by stratum, there is two polynomials p, q in R[V ] such that f = p/q is regular with restriction to S 1 . In particular γ(q) = 0 and we get γ(f ) = γ(p) γ(q) . The condition γ(f ) / ∈ R[[t 1/ N ]] says that ord t (γ(p)) < ord t (γ(q)). By the Artin approximation theorem ( [START_REF] Artin | On the solutions of analytic equations[END_REF]), there exists δ : R[V ] → R[[t 1/ N ]] alg that approximate γ as close as desired for the (t)-adic topology. More precisely, if S 1 = {r = 0, s = 0}, where r, s are two polynomials, one may choose δ satisfying δ(r) = 0, ord t (γ(p)) = ord t (δ(p)), ord t (γ(q)) = ord t (δ(q)), ord t (γ(s)) = ord t (δ(s)).

Then, one deduces that ord t (δ(p)) < ord t (δ(q)) and δ ∈ S 1 . Thus, δ(f ) = δ(p) δ(q) . By theorem 3.5, one knows that δ(f ) ∈ R[[t 1/ N ]] alg and hence ord t (γ(p)) ≥ ord t (γ(q)), a contradiction. This concludes the proof.

Remark 3.8. The proof of Theorem 3.7 heavily relies on the existence of a stratification, associated with a given continuous rational function, such that the function is regular on the strata. This result can also be generalized (with exactly the same proof) to a singular real algebraic variety if we replace the ring of continuous rational functions by the ring of regulous functions as considered in Theorem 2.13. Moreover, all the previous results that concern nonsingular real algebraic varieties can be extended to central real algebraic varieties with isolated singularities since in this case rational continuous functions are still regulous [START_REF] Monnier | Semi-algebraic geometry with rational continuous functions[END_REF]Theorem 2.25]. However, the question to know whether we get the same notion if we replace algebraic Puiseux arcs with formal ones in the definition of the substitution along arcs for general rings (definition 3.1) seems open.

Another natural question which arises is what can be said about substitution along arcs for continuous rational functions defined on singular varieties. Note already that this times considering Puiseux arcs or formal arcs makes a crucial difference.

Example 3.9. Consider the (singular but central) variety V ⊂ R 3 defined by the equation y 3 = z 2 x 3 . The arc γ given by x = y = 0 and z = t is contained in V . Let us now consider the rational function f = y/x on V . This function is continuous on V because of the relation

f 3 = z 2 . Note now that the power series arc γ ∈ (R[[t]] alg ) 3 lifts to f • γ = t 2/3 which belongs to R[[t 1/ N ]] alg but not to R[[t]] alg .

Substitution property for singular varieties

We give some examples and partial results under some assumptions and perform also a counterexample showing that there is no hope to get the substitution property along arcs for continuous rational functions defined on a general singular variety. 4.1. First facts. From now on, let V ⊂ R n be a singular irreducible central algebraic variety. We first state an elementary result that will imply the weak substitution property. Proposition 4.1. Let A = R[V ] the coordinate ring of V ⊂ R n and B an A-algebra which is a subring of the ring of semi-algebraic continuous functions on V . Assume that B satisfies the Łojasiewicz property over A. Namely for any f and g in B such that Z(f ) ⊂ Z(g), there is h ∈ B and an integer n such that g n = f h.

Then, B satisfies the weak substitution property over points.

Proof. Let φ : A → R be a morphism. Since B is a subring of the ring of semi-algebraic continuous functions, φ admits a canonical lifting to B. Hence, one has only to show the unicity. The argument (already met in the proof of theorem 2.12) is the following. One may assume, for simplicity, that φ(x 1 ) = . . . = φ(x n ) = 0. Let us assume then, for simplicity, that the origin

o of R n is in V . Let f ∈ B. Up to considering f -f (o), one may assume that f (o) = 0.
Then, by Łojasiewicz property applied to x 2 1 + . . . + x 2 n and f , there exists an integer N and a semi-algebraic continuous function g ∈ B such that f N = (x 2 1 + . . . + x 2 n )g. This algebraic identity implies that φ(f ) = 0 which concludes the proof.

As a consequence, since the ring R 0 (V ) has the Łojasiewicz property, one gets : Proposition 4.2. The ring R 0 (V ) of continuous rational functions on V satisfies the weak substitution property on points.

One can get also the substitution along arcs but under some additional hypothesis. Proposition 4.3. Let π : V → V be a birational regular morphism with V non-singular, such that π is an homeomorphism for the Euclidean topology. Then R 0 (V ) is isomorphic to R 0 ( V ) and satisfies the substitution property along arcs.

Proof. The canonical morphism R 0 (V ) → R 0 ( V ), given by the composition by π, is an isomorphism whose inverse isomorphism is given by the composition by the continuous rational map π -1 (the composition is still rational continuous because π is birational).

We know, by theorem 3.5, that R 0 ( V ) satisfies the substitution property along arcs, since V is non-singular. Hence, the substitution property along arcs holds also for R 0 (V ). One may illustrate this result with the following example.

Example 4.4. Let us consider the surface with equation x 3 = (1 + z 2 )y 3 , for which continuous rational functions and regulous functions differ [START_REF] Kollár | Continuous rational functions on real and p-adic varieties[END_REF]. The blowing-up V → V along the z-axis is an homeomorphism and hence the ring R 0 (V ) does satisfy the substitution property along arcs. Note moreover that for the continuous rational function x/y, the denominator has a zero set of codimension only one (to be compared with the smooth case given by Proposition 2.9).

One may generalize a bit the previous proposition in a version which will be useful in the next subsection. Proposition 4.5. Let W ⊂ R m be an irreducible real algebraic variety, π : W → V a surjective regular birational morphism and assume V is central. Then, R 0 (V ) is isomorphic to the ring R π 0 (W ) of continuous rational functions on W which are constant on the fibers of π. Proof. Let us consider the morphism π * : R 0 (V ) → R π 0 (W ) defined by f → f • π. This morphism is well defined since, being a composition of continuous functions, the function f • π is also continuous, and also constant on the fibers of π by construction. It is moreover rational since π is birational. And the morphism π * is injective since π est surjective.

Finally, π * is surjective since any continuous function on W constant on the fibers of π goes down to a continuous function on V . By birationality of π, this new function on V remains rational if the function on W was so. Remark 4.6. Note that if V is a central variety and π : W → V is a blowing-up, then π is surjective and one may use proposition 4.5.

4.2.

A counterexample. The aim of this subsection is to show that the ring R 0 (V ) does not necessarily satisfy the substitution property along arcs if V is singular. Theorem 4.7. Let us denote by V the central hypersurface in R 4 given by the equation

x 8 = (z 2 1 + z 2
2 )y 8 . The ring R 0 (V ) of continuous rational functions on V does not satisfy the substitution property along arcs.

Proof. We are going to prove that the unicity in the substitution fails. First note that the variety V is central, irreducible and singular along the plane of coordinates (z 1 , z 2 ) and also along the y axis.

Let us remark that the rational function x 2 /y 2 , which is well defined outside the singular plane, does extend by continuity on the whole V to the function 4 z 2 1 + z 2 2 , and hence x 2 /y 2 defines a function g in R 0 (V ). Note that it is not the case for the function x/y. Indeed, for all (x, y, 1, 0) ∈ V , one has x y 8 = 1 and hence x y → ±1 as (x, y) → (0, 0). Since x and y can take positive and negative values on X, x y does not admit a limit as (x, y) goes to (0, 0). The blowing-up π : W → V of V along the plane of coordinates (z 1 , z 2 ) can be seen in the chart given by the coordinates

x = uv, y = v, z 1 = w 1 , z 2 = w 2
as the variety with equation u 8 = w 2 1 + w 2 2 . Hence W is the product of a surface S in R 3 , defined by the same equation as the one of W , with a line (the v-axis). The surface S has an isolated singularity at the origin, hence the singular locus of W consists of the v-axis.

Since V is a central variety, the blowing-up π is surjective (see remark 4.6) and, using 4.5, one may identify R 0 (V ) with the ring R π 0 (W ) of continuous rational functions on W which are constant on the fibers of π.

We are going to exhibit an algebraic Puiseux arc contained in the variety V which can be lifted to the ring R 0 (V ) in two different ways. Let us denote by γ : [0, ǫ[→ V the polynomial arc defined by γ(t) = (0, 0, 0, t). The arc γ is contained in the singular plane of V . Moreover, it can be lifted to W into exactly two different algebraic Puiseux arcs given by

γ± (t) = (± 4 √ t, 0, 0, t). Lemma 4.8. Let f ∈ R 0 (V ). The semi-algebraic function t → f • γ(t)
is the composition of a one variable rational function F , regular at any point in [0, +∞), with the square root function, i.e.

(f • γ)(t) = U ( √ t) V ( √ t)
where U and V are one-variable polynomials such that V (0) = 0. Proof. It suffices to show the result for the function f • π composed with a lifting of the arc γ on Y (for instance γ+ ) since f

• γ = f • π • γ+ .
We want to restrict the function f • π, which is continuous rational on W , to the image of γ+ . The Zariski closure of the image of γ+ is the nonsingular irreducible curve C defined by u 4 = w 2 and v = w 1 = 0 in S × {0} ⊂ W . Note that C is not contained in the singular locus of W . Therefore the restriction of f • π to C is still rational by [START_REF] Kollár | Continuous rational functions on real and p-adic varieties[END_REF]Proposition 8], and of course continuous. In particular, it is a regular function on C because C is a nonsingular curve.

Parametrizing C with θ → (θ, θ 4 ), one gets the existence of a regular function G on R such that f • π(θ, 0, 0, θ 4 ) = G(θ). Moreover, this regular function is even since (θ, 0, 0, θ 4 ) ∈ W and (-θ, 0, 0, θ 4 ) ∈ W lies over the same point (0, 0, 0, θ 4 ) ∈ V and f ∈ R 0 π (W ). Hence, we claim there is a one-variable rational function F , regular at any point in [0, +∞), such that G(θ) = F (θ 2 ). Indeed, we may write G(θ) = R(θ)/(1 + Q(θ)) with R, Q some real onevariable polynomials and such that Q is a sum of squares of one-variable polynomials. Since G(θ) = (R(θ)(1 + Q(-θ)))/((1 + Q(θ))(1 + Q(-θ))) and ((1 + Q(θ))(1 + Q(-θ))) have even parity then it follows that (R(θ)(1 + Q(-θ))) has also even parity and we get the claim. To end, one sets θ 4 = t and gets

(f • γ)(t) = G( 4 √ t) = F ( √ t)
as desired.

Lemma 4.9. Any morphism φ : R 0 (V ) → R[[t 1/ N ]] alg which extends the morphism given by the evaluation along γ on polynomial functions, is entirely determined by the image of the continuous rational function g (which extends x 2 /y 2 by continuity on all V ).

Proof. According to lemma 4.8 and since (g • γ)(t) = √ t, for any f ∈ R 0 (V ), there is a onevariable rational function F regular at any point in [0, +∞) such that f • γ = F (g • γ). Hence, for any t ∈ [0, ǫ[, one has (f -F • g)(γ(t)) = 0. Note also that F • g ∈ R 0 (V ). Since the image of γ is locally contained in the zero set of x 2 + y 2 + z 2 1 , one has, by Łojaciewicz equality ([2, théorème 2.6.6]), the existence of an integer N and a semi-algebraic continuous function h such that (f -F • g) N = (x 2 + y 2 + z 2 1 )h. Note that this equality shows also that h ∈ R 0 (V ). Hence, for any given morphism φ, one has φ((f -F • g) N ) = 0 which gives φ(f ) = φ(F • g). Since F is regular at 0, one may evaluate its denominator at any arc going through the origin, and hence one gets also φ(f ) = F (φ(g)).

In particular, one deduces that there exist at most two candidates for such morphisms : either φ + which sends g onto √ t or φ -which sends g onto -√ t. The morphism φ + is the natural morphism coming from the evaluation of continuous semi-algebraic functions along γ. The next lemma states that the choice φ -is also possible. Proof. One has to check that φ -does respect algebraic relations existing for elements of R 0 (V ) over R[V ], the ring of polynomial functions over V . Let f 1 , . . . , f n in R 0 (V ) such that P (f 1 , . . . , f n ) = 0 where P ∈ R[V ]. One has to show that φ -(P (f 1 , . . . , f n )) = 0.

According to the end of the proof of the previous lemma, there exists a one-variable rational function F , regular at 0, such that φ + (P (f 1 , . . . , f n )) = F (φ + (g)) and φ -(P (f 1 , . . . , f n )) = F (φ -(g)).

Let us write F = U/V where U and V are one-variable polynomials.

Since the natural morphism φ + is well defined, one gets from P (f 1 , . . . , f n ) = 0 that φ + (P (f 1 , . . . , f n )) = 0. Namely, one has F (φ + (g)) = 0, which means that U(φ + (g)) = 0.

One may write U(x) = S(x 2 ) + xT (x 2 ) where S and T are one-variable polynomials. Then, U(t) = 0 for any t ∈ [0, √ ǫ[, and hence S = T = 0, i.e. U = 0.

Consequently, F (φ -(g)) = 0 and hence φ -(P (f 1 , . . . , f n )) = 0, the desired equality.

As a consequence the unicity in the substitution property fails.

2. 1 .

 1 Definition and first examples.

  Indeed, one has to show that any arc γ : R[V ] → R, where R = R[[t 1/ N ]] alg or R[[t]] alg , admits a unique lifting to R[ V ]. One may compose γ with the evaluation morphism e u : R → R sending t onto u ∈ [0, ǫ[. Since V → V is a bijection, the morphism e u • γ admits a unique lifting to R[ V ] for any u ∈ [0, ǫ[. This shows that γ has a unique lifting to R[ V ].

Lemma 4 . 10 .

 410 The setting φ -(g) = -√ t defines a morphism φ -: R 0 (V ) → R[[t 1/ N ]] alg which extends the morphism given by γ on R[V ] the ring of polynomial functions over V .

We thank Michel Coste for fruitful discussions on the topic.