In this work we prove some inequalities for smooth algebraic functions

(smooth solutions to polynomial equations) which are crucial for proving some scaling

properties of their averages and maxima, that are typical in the case of polynomials. As

a byproduct, it is shown that x !−→ (y −f(x))2, where f is a smooth algebraic function,

behaves like a polynomial (in terms of scaling properties of averages and maxima)

A. PARMEGGIANI

Directory of Open Access Journals

Rendiconti di Matematica, Serie VIIVolume 15, Roma (1995), 25-38On Some Uniform Boundsfor Smooth Algebraic FunctionsA. PARMEGGIANIRiassunto: In questo lavoro si dimostrano alcune disuguaglianze relative a fun-zioni algebriche C" (cioè soluzioni C" di equazioni polinomiali) che sono cruciali perprovare proprietà di scala di medie e massimi delle suddette funzioni, tipiche nel casopolinomiale. Si ottiene inoltre che x !"# (y " f(x))2, dove f è una funzione algebricaliscia, si comporta come un polinomio (relativamente alle proprietà di scala di medie emassimi).Abstract: In this work we prove some inequalities for smooth algebraic functions(smooth solutions to polynomial equations) which are crucial for proving some scalingproperties of their averages and maxima, that are typical in the case of polynomials. Asa byproduct, it is shown that x !"# (y " f(x))2, where f is a smooth algebraic function,behaves like a polynomial (in terms of scaling properties of averages and maxima).1 – IntroductionThe purpose of this paper is to establish some polynomial-like prop-erties of smooth real-valued algebraic functions, i.e. smooth solutions topolynomial equations. The properties we are interested in are similar tothose stated in Fe!erman [1]:if P (x) is a polynomial of degree # d then, with Q a (closed) cube of IRnKey Words and Phrases: Bernstein’s inequalities – Polynomial equationsA.M.S. Classification: 26D10 – 26B35 – 14P9926 A. PARMEGGIANI [2]and constants depending only on d and the dimension n :Avx$Q|P | # maxx$Q|P | # CAvx$Q|P |;(a)maxx$Q|5P | # C(diamQ)!1 maxx$Q|P |;(b)If P ! 0 on Q then - a subcube Q# . Q with (diamQ#) ! c(diamQ)on which(c) minx$Q!P ! 12maxx$QP.(Notice that (c) is a consequence of (b).) We shall refer to these propertiesas polynomial-like scaling properties.The work of Stein and his collaborators (see for instance Nagel-Stein-Wainger [9]) brought to light that a subelliptic di!erential operator is gov-erned by a family of non-Euclidean balls. In Parmeggiani [10] and [11] afamily of non-Euclidean balls in the cotangent bundle of IRn is attachedto the (total) symbol p(x, &), supposed nonnegative, of a subelliptic pseu-dodi!erential operator, by embedding the unit cube through canonicaltransformations satisfying suitable estimates on the derivatives(1). A cru-cial step in this construction is an extension of the above properties (a),(b) and (c) (and a few more) to smooth real-valued algebraic functionsand to polynomials evaluated on graphs of smooth algebraic functions.These results have been generalized by C.Fe!erman and R.Narasimhanin [6] and [7], works in which they prove also similar properties for poly-nomials evaluated on higher codimensional smooth algebraic varieties.We start by proving some ”ellipticity” properties of the average, withrespect to one of the variables, of a nonnegative polynomial. We thenprove two theorems about scaling properties of averages and maxima offunctions whose gradients are ”controlled” by the function itself ( in termsof L"$norm). Afterwards we show that the aforementioned propertiesextend to smooth algebraic functions and to polynomials of the kind(y $ X)d, where y " IR is a parameter and X is a real variable, whenevaluated at X = f(x) with f a smooth algebraic function, and when(1)This results in necessary and su"cient conditions for L2"a-priori bounds for subel-liptic operators. See Fe!erman [1], Fe!erman-Phong [2,3,4,5].[3] On Some Uniform Bounds etc. 27evaluated at X = f(x1, x#) $ (Avx1f)(x#), with f a smooth algebraicfunction in x1, polynomial (of a-priori bounded degree) in x#. Looselyspeaking, one has to study that in order to understand the geometry ofone of the main ”normal forms” (after a symplectomorphism) of p(x, &)on a box of fixed size:p(x, &) = &21 + (&2 $ -(x1, x2))2 + V (x1, x2) ,V (x1, x2) = p(x1, x2, 0, -(x1, x2))where, upon rescaling to the unit cube, - is a smooth algebraic functionin x1, polynomial in x2, and p(x1, x2, 0, &2) is a polynomial, both of a-priori bounded degree and maximum norms. Here we study the estimatesrelative to the ”quadratic” part of p. The much more di%cult case of theestimates relative to V are treated in [6] and [7].We address the interested reader to [10] and [11] for more detailsabout the use of these polynomial-like properties.It should be stressed once more the novelty here is that also in thecase of smooth algebraic functions we have a complete control on the sizeof the regions in terms of the size of the functions (when one wants to getinformations about maxima and averages; a typical example is property(c) above), and on the scaling properties of the L"$norms in terms ofthe sizes of the regions on which the norms are taken.2 – The Results.We start by studying some scaling properties of averages, with respectto one of the variables, of polynomials and of maxima of smooth functionswith ”controlled” gradient(2).Proposition 2.1. Suppose 0 # f(x1, x2) is a nonnegative poly-nomial of degree d for (x1, x2) " IR 6 IRN . Take (x1, x2) " I 6 Q, withdiamI 7 ., diamQ 7 /, 0 < / < 1, / # . # 1, and take x02 " Q. Suppose,Avx1$If-(x2) 7 /4 )x2 " Q( (the double of Q, as usual). Then there(2)In the sequel, every constant C, c, c(n, d), c1, c2, c3, c4, c5, C!, is a universal constant.For A, B $ 0, A % B means that &C, c > 0, universal constants, such that cB ' A 'CB.28 A. PARMEGGIANI [4]exist Q1 . Q, I1 . I, center(Q1) = x02, diamQ1 7 / 7 diamI1, such thatf(x1, x2) 7 /4, )(x1, x2) " I1 6 Q1 .Proof. We have,f(x01, x02) = maxx1$If(x1, x02) ! c1/4(since f is a nonnegative polynomial and,Avx1$If-(x02) 7 /4) for somex01 " I and max(x1,x2)$I)Q f(x1, x2) 7 /4.Choose now Ĩ1, x01 " Ĩ1, with diamĨ1 = c0/. Since Ĩ1 . I, maxĨ1)Q f #c2/4.Also, with a universal constant c = c(N +1, d), f being a polynomial! 0,maxĨ1)Q|5f | # cc2//4 = c3/3 .We can then find I1 . Ĩ1, Q1 . Q, Q1 centered at x02, with diamI1 =c4/ = diamQ1, so that diam8I1 6 Q19 7 diam8Ĩ1 6 Q9andc5 maxĨ1)Qf # minI1)Q1ffor a universal constant c5 : for (x1, x2) " I1 6 Q1,f(x1, x2) = f(x01, x02) +! 10<85f98x0 + t(x $ x0)9, (x $ x0)> dt !! f(x01, x02) $ 2c4c3/4 ! (c1 $ 2c4c3)/4 =, c12c2-c2/4 ! c5 maxI1)Q1fwhen c4 = c1/(4c3) and c5 = c1/(2c2).The meaning of the above Proposition is that if the average of anonnegative polynomial with respect to one of the variables is ”elliptic”,then, in a smaller box (whose size we have control on), the polynomial is”elliptic” in all the variables.Theorem 2.2. Let Q . IRn be a (closed) cube and Q( = 2Q. Letf " C"(Q() be such that(5f(L"(Q) #cdiamQ(f(L"(Q) .[5] On Some Uniform Bounds etc. 29Then(1) Avx$Q|f | 7 maxx$Q|f |(i.e., as always, the two quantities are equivalent by universal constantsindependent of f, depending only on n and c).Proof. We can suppose Q to be the unit cube centered at the origin.-x̄ " Q such that (f(L"(Q) = |f(x̄)|.Recall that if diamQ = 0 then |Q| = |side(Q)|n = (0/8n)n.So, choose a cube Q1 . Q with x̄ " Q1 and diamQ1 = 1/(2c̃), wherec̃ = max{1, c}. Hence |Q1| 7 |Q|. We have,f(x) = f(x̄)+ <! 1085f98x̄ + t(x $ x̄)9dt, (x $ x̄)>== f(x̄)+ <F (x, x̄), (x $ x̄)> .ThenAvx$Q|f | !|Q1||Q| Avx$Q1 |f | ! c+++ 1|Q1|!Q1f(x)dx+++ == c+++ 1|Q1|!Q11f(x̄)+ <F (x, x̄), (x $ x̄)>2dx+++ !! c1|f(x̄)| $ 12(f(L"(Q)2=12c(f(L"(Q) ,since!Q1| <F (x, x̄), (x $ x̄)> |dx # c(f(L"(Q),diamQ1-|Q1| ,whencec(f(L"(Q) # Avx$Q|f | # (f(L"(Q) .Theorem 2.3. Suppose f " C"(Q(0) and )Q . Q0 (all the cubesconsidered are closed cubes)(5f(L"(Q) #cdiamQ(f(L"(Q) .30 A. PARMEGGIANI [6]Suppose diamQ 7 diamQ0, then(2) (f(L"(Q) 7 (f(L"(Q0).Proof. As usual, we can suppose Q0 to be the unit cube in IRncentered at the origin. Let x0 := x00 " Q0 be such that |f(x0)| =maxx$Q0 |f(x)|. Set c̃ = max{1, c}.If x0 " Q we are done.Suppose x0 *" Q. Let xQ be the center of Q and let L0 be the linethrough x0 and xQ and consider, on L0, the segment [x0, xQ]L0 . LetF #1(Q), F##1 (Q) be two parallel faces of Q, opposite with respect to xQ,at which L0 meets Q transversally, such that the point F#1 2 L0 is closerto x0 along L0, than the point F##1 (Q) 2 L0.(In case L0 intersects Q in a corner or vertex, we just choose one ofthe possible faces).On [x0, xQ]L0 choose x1 such that distL0(x0, x1) = 1/(2c̃), wheredistL0 is the distance on the line L0.We have that F ##1 (Q) . H1, a hyperplane. Choose H(x1) to be thehyperplane parallel to H1 through x1. By convexity of Q0, the segment ofL0, [x0, L0 2 F ##1 (Q)]L0 . Q0, and )t " [x0, L0 2 F ##1 (Q)]L0 , H(t) 2 Q0 *= 3,where H(t) is the hyperplane parallel to H1 through t. Denote by P theband between the boundaries H(x1) and H1. Then Q . P 2 Q0.Notice that side(Q) # dist(H(x1), H1) < side(Q0).Hence there exists Q1 with F##1 (Q) . F ##1 (Q1) . H1 and x1 " 1Q1, sothat Q . Q1 . P 2 Q0. Let x01 " Q1 be such that (f(L"(Q1) = |f(x01)|. Ifx01 " Q we stop here, otherwise consider the line L1 through x01 and xQ,and a point x2 " [x01, xQ]L1 with distL1(x01, x2) = diamQ1/(2c̃).Consider now, with obvious notations, H(x2) parallel to H2. ThenQ . P1 2 Q1 . P 2 Q0 and side(Q) # dist(H(x2), H2) < side(Q1).Therefore there exists Q2, x2 " 1Q2, F ##2 (Q) . F ##2 (Q2) . H2, andQ . Q2 . Q1 2 P1, F ##2 (Q) being the farthest face of Q, along L1, withrespect to x01.Notice that diamQ # diamQ2 # Q0.Suppose we constructed the Qj’s, j = 0, 1, 2, . . . , k (so, in particular,x0j *" Q, )j, x0j a point of maximum for |f | on Qj), we want to constructQk+1.[7] On Some Uniform Bounds etc. 31Recall that, for j = 1, 2, . . . , k, we haveQ . Qj . Qj!1 2 Pj!1(Pj’s are determined by pairs of hyperplanes parallel to the coordinate-hyperplanes. Notice that Pj 2 Qj is an n$dimensional parallelepiped).Consider x0k " Qk such that (f(L"(Qk) = |f(x0k)|.If x0k " Q we stop here the construction of the sequence of cubes,otherwise let Lk be the line through x0k and xQ. Take xk+1 " [x0k, xQ]LkwithdistLk(x0k, xk+1) =diamQk2c̃.Then, with the obvious notations, consider H(xk+1) and Hk+1 (chosen asabove). Then Q . Pk 2 Qk . Pk!1 2 Qk!1. Againside(Q) # dist(H(xk+1), Hk+1) < side(Qk) =9=9 -Qk+1, F ##k+1(Q) . Fk+1(Qk+1) . Hk+1such that xk+1 " 1Qk+1, Q . Qk+1 . Qk 2 Pk.Notice that, )j, diamQ # diamQj # Q0.Since, at each step, we shrink the region by an amount > diamQ/(2c̃),there exists N such that N # 2c̃(diamQ0/diamQ), and the constructionstops at xN+1, i.e. xN+1 " Q.Then we have, )j, j = 0, 1, . . . , N :f(x0j) = f(xj+1)+ <F (x0j , xj+1), (x0j $ xj+1)>,so that,(f(L"(Qj) # (f(L"(Qj+1) +12(f(L"(Qj),i.e.(f(L"(Qj) # 2(f(L"(Qj+1),whence(f(L"(Q0) # 2N(f(L"(Q) .Our aim is now to show that algebraic functions, i.e. solutions to apolynomial equation, satisfy the hypotheses of Theorem 2.2 and 2.3 (andhence enjoy a polynomial-like scaling property).32 A. PARMEGGIANI [8]Theorem 2.4. Let Q = Q1 6 I be the unit cube, centered atthe origin, in IRn+1, with coordinates (x, y) " IRn 6 IR. Let P (x, y) be apolynomial of a-priori bounded degree d, with |1yP | ! C > 0, )(x, y) "Q(, and (P(L"(Q#) # C(, for fixed constants C, C( > 0. Let y = f(x) bethe solution to P (x, y) = 0 on Q(, with f " C"( 12Q(1), (f(L"(Q1) # 2.Then, for |0| # 2 (and actually )0)(3) (1'x f(L"(Q1) # C'(P(L"(Q#)8M $ m9 # C(C'8M $ m9where M = maxx$Q1 f(x), m = minx$Q1 f(x) and the C'’s depend onlyon n + 1 and d.Proof. Notice that, by hypothesis, J = [m, M ] . I(. We haveP (x, y) =! 10(1yP )8x, f(x) + t(y $ f(x))9dt,y $ f(x)-so that, )x " Q1, )y " J, P being a polynomial,max(x,y)$Q1)J|P (x, y)| # c(d, n) max(x,y)$Q1)J|y $ f(x)| = c(d, n)|M $ m| .It follows that, )y " J,|1xP (x, y)| # c(d, n) maxx$Q1|P (x, y)| ## c(d, n)(P(L"(Q#)|M $ m| # c(d, n)C(|M $ m| ,so that,max(x,y)$Q1)J|1xP (x, y)| # c(d, n)(P(L"(Q#)|M $ m| # c(d, n)C(|M $ m| .Hence, since |(1xP )8x, f(x)9| # max(x,y)$Q1)J |1xP (x, y)|, we obtain, us-ing the formula from the Implicit Function Theorem, for |0| = 1,1'x f(x) = $(1'x P )8x, f(x)9(1yP )8x, f(x)9 ,[9] On Some Uniform Bounds etc. 33that, |0| = 1,(1'x f(L"(Q1) #c(d, n)C(P(L"(Q#)|M $ m| #c(d, n)CC(|M $ m| .Now, for |0| = 2, 0 = 01 + 02, |01| = |02| = 1,1'x f(x) = $1(1'x P )8x, f(x)9+ (1'1x 1yP )8x, f(x)91'2x f(x)(1yP )8x, f(x)9 +$ (1'1x P )8x, f(x)9(1'2x 1yP )8x, f(x)9+ (12yP )8x, f(x)91'2x f(x)(1yP )8x, f(x)922.As we already know, |(1'1x P )8x, f(x)9| # C(M $ m).For (1'x P )8x, f(x)9we have: )y " J|1'x P (x, y)| # maxx$Q1|1'1x P (x, y)| # C(M $ m),whence max(x,y)$Q1)J |1'x P (x, y)| # C(M $ m) and|(1'x P )8x, f(x)9| # max(x,y)$Q1)J|1'x P (x, y)| # C(M $ m),with, clearly, C = c(d, n).Now,|(1'2x 1yP )8x, f(x)9| # max(x,y)$Q#|1'2x 1yP (x, y)| ## c(d, n)(P(L"(Q#) # c(d, n)C( ,|(12yP )8x, f(x)9| # max(x,y)$Q#|1(yP (x, y)| ## c(d, n)(P(L"(Q#) # c(d, n)C( .It follows, for |0| = 2,(1'x f(L"(Q1) # c(d, n)(P(L"(Q#)|M $ m| # c(d, n)C(|M $ m|34 A. PARMEGGIANI [10]Corollary 2.5. Same hypotheses as in Theorem 2.4. )Q#1 . Q1,we have, for |0| # 2,(4) (1'x f(L"(Q!1) #c(d, n)C((diamQ#1)|'|,maxx$Q!1f(x) $ minx$Q!1f(x)-.Proof. The proof follows from the proof of Theorem 2.4 consideringJ(Q#1) = [minQ!1f,maxQ!1f ] = [m(Q#1), M(Q#1)],and noticing the following facts: )y " J(Q#1), )x " Q#1,|(1xP )8x, f(x)9| # c(d, n)diamQ#1C(|M(Q#1) $ m(Q#1)|,since, )y " J(Q#1), )0 :maxx$Q!1|1'x P (x, y)| # c(d, n) maxx$Q!1|P (x, y)|(diamQ#1)!|'| ## c(d, n)(P(L"(Q#)(diamQ#1)|'||M(Q#1) $ m(Q#1)| ## c(d, n)C((diamQ#1)|'||M(Q#1) $ m(Q#1)|and|(1'y P )8x, f(x)9| # max(x,y)$Q#|1'y P (x, y)| ## c(d, n)(P(L"(Q#) # c(d, n)C( .A very nice consequence is the following:Corollary 2.6. Same hypotheses as in Theorem 2.4. Then:(5) (5f(L"(Q) 7,maxx$Qf(x) $ minx$Qf(x)-.[11] On Some Uniform Bounds etc. 35This allows us to prove theCorollary 2.7. Let f be a smooth algebraic function satisfying theconditions in Corollary 2.6. Consider, for fixed y " IR, the polynomial inX " IR, Py(X) = (y$X)2, and the associate function py(x) = (y$f(x))2.Then:(6) Avx$Qpy(x) 7 maxx$Qpy(x) ,and(7) (1xpy(L"(Q) # C(py(L"(Q) ,where C and the constants in the equivalence do not depend on y.Proof. Let J = [minx$Q f,maxx$Q f ]. Consider 1XPy(X) = $2(y $X). Recall that diamQ 7 1. Then, for a universal constant C (as usualall the constants C are universal constants),maxX$J|1XPy(X)| #C|J | maxX$J |Py(X)| .Hence, since: x " Q =9 f(x) " J,|1xpy(x)| # 2 maxX$J|1XPy(X)|(1xf(L"(Q) ## 2C|J | maxX$J |Py(X)|(1xf(L"(Q) ## C#|J |,maxQf $ minQf-(Py(L"(I) = C # maxx$Q|py(x)| ,whence(1xpy(L"(Q) # C(py(L"(Q) , ,andAvx$Qpy 7 (py(L"(Q) .Remark 2.8. Of course the Corollary holds true for Py(X) = (y $X)d, d ! 1. We stated it for d = 2 since this is what is needed in [10] and[11].36 A. PARMEGGIANI [12]Given now the algebraic function f(x1, x2), we have to examine thescaling properties of g(x1, x2) = f(x1, x2) $ (Avx1$If)(x2) := f(x1, x2) $f̄(x2).Lemma 2.9. Suppose f, P satisfy the hypotheses of Theorem 2.4.Suppose now Q = Q1 6 I = I 6 Q2 6 I, Q2 the unit cube in IRn!1,(x1, x2) " I 6 Q2. Define f̄(x2) = (Avx1$If)(x2). Then, for a constant Cindependent of f and x2, we have, )x2 fixed,(1x1g(., x2)(L"(I) # C(maxx1$I g(x1, x2) $ minx1$I g(x1, x2));(i)(1x18g(., x2)29(L"(I) # C(g(., x2)2(L"(I);(ii)(1x1g(L"(Q) # C(g(L"(Q) .(iii)Proof. DefineM(g)(x2) = maxx1$Ig(x1, x2) =,maxx1$If(x1, x2)-$f̄(x2) = M(f)(x2)$f̄(x2),and, with m(f)(x2) = minx1$I f(x1, x2),m(g)(x2) = minx1$Ig(x1, x2) = m(f)(x2) $ f̄(x2) .)x2 " Q2 fixed, consider J(x2) = [m(f)(x2), M(f)(x2)]. Then|P (x1, x2, y)| # C(P(L"(Q#) max(x1,y)$I)J(x2)|y $ f(x1, x2)| ## C(P(L"(Q#)++M(f)(x2) $ m(f)(x2)++ ,C independent of x2. It follows that|1x1f(x1, x2)| # C|(1x1P )8x1, x2, f(x1, x2)9| ## C(P(L"(Q#)++M(f)(x2) $ m(f)(x2)++ ,being diamI 7 1. SinceM(g)(x2) $ m(g)(x2) = M(f)(x2) $ f̄(x2) $ m(f)(x2) + f̄(x2) == M(f)(x2) $ m(f)(x2)[13] On Some Uniform Bounds etc. 37and 1x1g(x1, x2) = 1x1f(x1, x2), point (i) and (iii) follow at once.About (ii) :1x18g(x1, x2)29= 2g(x1, x2)1x1g(x1, x2),therefore|1x18g(x1, x2)29| # 2C(g(., x2)(2L"(I) = 2C(g(., x2)2(L"(I) .Corollary 2.10. Under the same hypotheses, suppose further thatf(x1, x2) is a polynomial of bounded degree D in x2. Then the Bernstein’sinequality holds for g(x1, x2) := f(x1, x2) $ f̄(x2) :(8) (5g(L"(Q) # C(g(L"(Q) .Corollary 2.11. Same hypotheses of Corollary 2.10. Considerthe functionpy(x1, x2) =8y $ g(x1, x2)92.Then(9) Avx$Qpy 7 maxx$Qpy(x)and(10) (1xpy(L"(Q) # C(py(L"(Q) ,for universal constants independent of y.38 A. PARMEGGIANI [14]REFERENCES[1] C.L. Fefferman: The Uncertainty Principle, Bull. of A.M.S. 9,, No.2 (1983).[2] C.L. Fefferman – D.H. Phong: The Uncertainty Principle and Sharp G̊ardingInequalities, Comm. Pure Appl. Math. 34 (1981).[3] C.L. Fefferman – D.H. Phong: On the Asymptotic Eigenvalue Distribution ofa Pseudodi!erential Operator , Proc. Nat. Acad. Sci. U.S.A. 77 (1981).[4] C.L. Fefferman – D.H. Phong: Subelliptic Eigenvalue Problems, Conf. inHonor of A.Zygmund (W.Beckner et al., eds) Wadsworth Math. Series, 1981.[5] C.L. Fefferman – D.H. Phong: Symplectic Geometry and Positivity of Pseu-dodi!erential Operators, Proc. Nat. Acad. Sci. U.S.A. 79 (1983).[6] C.L. Fefferman – R. Narasimhan: Bernstein’s Inequality on Algebraic Curves,Preprint (1992).[7] C.L. Fefferman – R. Narasimhan: On the Polynomial-Like Behaviour of Cer-tain Algebraic Functions, (to appear).[8] C.L. Fefferman – A. Sanchez-Calle: Fundamental Solutions for Second OrderSubelliptic Operators, Annals of Math., 124 (1986).[9] A. Nagel – E.M. Stein – S. Wainger: Balls and Metrics Defined by VectorFields I: Basic Properties, Acta Math. 155 (1985).[10] A. Parmeggiani: Subunit Balls for Symbols of Pseudodi!erential Operators,Princeton Doctoral Dissertation (1993).[11] A. Parmeggiani: Subunit Balls for Symbols of Pseudodi!erential Operator , Pre-print, to appear in Advances in Mathematics.[12] A. Sanchez-Calle: Fundamental Solutions and the Geometry of the Sum ofSquares of Vector Fields, Invent. Math. 78 (1984).Lavoro pervenuto alla redazione il 13 luglio 1994ed accettato per la pubblicazione il 21 settembre 1994.Bozze licenziate il 21 ottobre 1994INDIRIZZO DELL’AUTORE:Alberto Parmeggiani – Dipartimento di Matematica – Università di Bologna – Piazza di PortaS.Donato 5 – 40127 Bologna – Italy

On Some Uniform Bounds for Smooth Algebraic Functions

https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/1995(1)/25-38.pdf

On Some Uniform Bounds for Smooth Algebraic Functions

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals