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Nearly-polynomial inverse theorem for the U^d norm in degree d+1

We prove a nearly polynomial inverse theorem for the Gowers $U^d$ norm, over finite fields of non-small characteristic, for polynomials of degree $d+1$. The case of degree $d$ was very recently settled by Milićević and Randelović with a fully polynom…

Authors: Tomer Milo, Guy Moshkovitz

Nearly-p olynomial in v erse theorem for the U d norm in degree d + 1 T omer Milo ∗ Guy Moshko vitz † Abstract W e pro ve a nearly p olynomial inv erse theorem for the Gow ers U d norm, o ver finite fields of non-small c haracteristic, for p olynomials of degree d + 1. The case of degree d was very recen tly settled by Mili ´ cevi ´ c and Rand ¯ elo vi´ c with a fully p olynomial b ound. W e moreov er pro vide a nearly p olynomial in verse theorem for homo gene ous p olynomials of an y degree smaller than 2 d . 1 In tro duction Let F = F p b e any prime finite field. The discr ete derivative of a function f : F n → F in direction v ∈ F n at the p oin t x ∈ F n is ∆ v f ( x ) = f ( x + v ) − f ( x ). More generally , the (discrete) deriv ative of order d , in directions v = ( v (1) , . . . , v ( d ) ) ∈ ( F n ) d at the p oin t x ∈ F n , is ∆ d v f ( x ) := ∆ v ( d ) · · · ∆ v (1) f ( x ) . A function f : F n → F is a p olynomial of degree b elo w d , denoted f ∈ Poly d , is that the deriv ative characterization is robust: f is c orr elate d with P oly 0 . In the represen tative case where g = e ( f ) with f : F n → F (e.g., Lemma 1.2 in [ TZ10 ]), the Go wers norm is simply ∥ g ∥ U d = bias(∆ d f ) 1 / 2 d , so Theorem 1.1 is a qualitative in verse to ( 1 ): ∃ Φ > 0 : cor 3 remains wide op en. 1.1 Go w ers inv erse in low degree Let GI d ( k ) denote the special case of the inv erse theorem for the Gow ers U d -norm ( 2 ) where the function f : F n → F is a p olynomial of degree at most k , and ch( F ) > k . V ery recen tly , in [ MR25 ] the first 3 case of the Polynomials Gow ers conjecture Inv erse w as pro ved, namely , GI d ( d ) with a p olynomial bound. This result relates to a separate line of w ork on the partition-v ersus-analytic rank conjecture for m ultilinear forms (e.g. [ Mil19 , Jan19 , CM22 , AKZ21 , CM23 , MZ22 , CY25 ]). Later, we explain in detail that connection, and how it relates to our work. In this paper w e obtain an almost-p olynomial b ound for GI d ( d + 1), the next op en case. With a sligh t abuse of notation, w e write x ˜ O d (1) for exp( − O d (log(1 /x ) 1+ o (1) )). (This is even stronger than a quasi-p olynomial bound, which has a constant O d (1) at the top instead of 1 + o (1).) 4 Theorem 1. L et d ≥ 1 . F or every f : F n → F with deg( f ) ≤ d + 1 , wher e d + 1 < ch( F ) , cor d , rk( g ) ≤ ˜ O d (AR( g )) . Mor e over, for every d -line ar form T , PR( T ) = ˜ O d (AR( T )) 6 One can naiv ely extend the definition of rk to non-homogeneous polynomials f ∈ Poly k b y defining rk( f ) := rk( f k ), but this is not useful in this pap er as this do es not give a decomposition of f , only of f k . 7 More precisely , w e can get ˜ O d ( x ) = O d ( x log( x + 1)). (In fact, w e can even get ˜ O d ( x ) = O d ( x (log | F | ( x + 1)+ 1)), whic h in particular is linear for mildly large F ). 4 2.2 Pro of outline W e start by sketc hing the standard proof of GI d ( d ). The argumen t is roughly as follows. F or f of degree d , the d -th deriv ative ∆ d f is a degree- d multilinear form, which is biased b y assumption. Call that bias δ . Using a partition-v ersus-analytic rank bound, such as Theorem 2.1 , one obtains a partition-rank decomp osition for ∆ d f of length log( 1 δ ) ˜ O d (1) . By plugging in the diagonal ( x, . . . , x ) and dividing by d !, we obtain a decomp osition of f d (the degree- d part of f ), meaning rk( f d ) ≤ log( 1 δ ) ˜ O d (1) . W riting f = P i α i β i , a standard F ourier/pigeonhole-type argument giv es a correlation for f d of at least δ ˜ O d (1) with some member of the span of the α i and β i —whic h is of degree at most d − 1—and thus a correlation for f itself with such a low er-degree p olynomial. The proof of Theorem 2 , for homogeneous polynomials, follows this classical recipe, except it utilizes a new correlation lemma (Lemma 3.1 ) to obtain a correlation with a linear combination of only the lo wer-degree factors from eac h pro duct α i β i in the decomp osition, which is a p olynomial of degree as low as deg( f ) / 2. The difficulty in the non-homogeneous case of Theore m 1 is tw ofold. First, ∆ d f is no longer homogeneous of degree d , so a short decomp osition of ∆ d f is not immediately a v ailable from its high bias. A significan t part of this pap er (Sections 4 and 5 ) is dedicated to the task of carefully obtaining a sp ecial decomp osition for a general f of degree d + 1 which—while not b eing necessarily a homogeneous decomp osition—can b e used to obtain a correlation with a p olynomial of degree at most deg ( f ) − 2. W e briefly sketc h now ho w this is done. In Section 4 , we in tro duce a new notion of rank for p olynomials called rk ∗ . The ob jectiv e here was to find a notion of rank for whic h the follo wing holds: whenever rk ∗ ( f ) is small, one can find a large correlation of f with a p olynomial of degree at most deg ( f ) − 2 (whereas for the usual notion of rank, one can only generally get deg( f ) − 1). It turns out that the requiremen t needed is quite w eak: if f = P r i =1 α i β i is a rk-decomp osition, for it to b e a rk ∗ decomp osition, w e require the degree-1 p olynomials in the decomp osition to ha ve linearly indep endent linear comp onen ts. This requiremen t guarantees that the zero-lo cus of these p olynomials is fairly large, which is essential for us to obtain significant correlation. W e now explain how the rk ∗ -decomp osition of f is obtained. W rite g = f d +1 and h = f d for the homogeneous parts of degree d + 1 and d resp ectiv ely . Proposition 5.1 provides a simple form ula for ∆ d f for an y p olynomial f of degree d + 1. Proposition 6.4 sho ws that decomposing g by the standard polarization process goes smoothly , thanks to the c hain rule for bias (Lemma 6.3 ), which tells us that ∆ d g is biased. The decomp osition of h is a more delicate issue, since the p olarization h need not b e biased. Ho wev er, w e pro ve that h is biased when adding an appropriate partial deriv ativ e ∂ c g ; moreo ver, the sum h + ∂ c g is a d -linear form, hence another application of Theorem 2.1 gives a decomp osition to this sum. T o deal with the summand ∂ c g w e added, we use a chain rule for rk ∗ (Prop osition 4.5 ). All of the ab ov e allo ws us to obtain a non-homogeneous decomp osition for f , whic h is nev ertheless useful for us since it is a short rk ∗ -decomp osition. Using the control w e hav e ov er its linear factors, Prop osition 4.7 allo ws us then to use this sp ecial decomposition to obtain significan t correlation with a p olynomial of degree deg( f ) − 2 ≤ d − 1, as needed. W e finally remark that we cannot utilize the arguments leading to the p olynomial b ound for GI d ( d ), in Theorem 3.4 , in order to obtain a similar b ound for degree d + 1, since our argument requires more con trol ov er the factors than is afforded b y that Theorem. 3 Correlation lemma, the homogeneous and the degree- d cases In this section w e prov e a correlation lemma which will b e imp ortan t in the rest of the pap er. As an almost immediate corollary , we will obtain Theorem 2 . 5 F or a tuple of functions A = ( A 1 , . . . , A m ), their joint zero set is Z( A ) = { x | ∀ i : A i ( x ) = 0 } . F or an ev ent E , w e abbreviate bias ( g | E ) = E [ χ ( g ) | E ]. Lemma 3.1 (a verage correlation lemma) . L et g, A 1 , . . . , A m : F n → F b e any functions, and put A = ( A 1 , . . . , A m ) . Then E c ∈ F m bias  g − m X i =1 c i A i  = P (Z( A )) · bias ( g | Z( A )) . Pr o of. By direct computation: E c ∈ F m bias  g − m X i =1 c i A i  = E x E c ∈ F m χ  g ( x ) − m X i =1 c i A i ( x )  = E x χ ( g ( x )) E c ∈ F m χ  m X i =1 c i A i ( x )  = E x χ ( g ( x )) 1 Z( A ) ( x ) = | F | − n X x ∈ Z( A ) χ ( g ( x )) = P (Z( A )) 1 | Z( A ) | X x ∈ Z( A ) χ ( g ( x )) = P (Z( A )) bias ( g | Z( A )) . W e will also need the following Theorem due to W arning (for a short pro of see [ Asg23 ]). Theorem 3.2 (W arning’s second theorem [ W ar35 ]) . F or al l p olynomials f 1 , . . . , f m : F n → F over a finite field F , if Z( f 1 , . . . , f m )  = ∅ then | Z( f 1 , . . . , f m ) | ≥ | F | n − P m i =1 deg( f i ) . W e obtain the following correlation b ound, where cor d ( g ) = max deg( P ) ≤ d cor( g , P ). Corollary 3.3. If Z( A ) ⊆ Z( g ) for some A = ( A 1 , . . . , A m ) ∈ F orm d ( F ) m , then cor d ( g ) ≥ | F | − dm . Pr o of. Apply Lemma 3.1 to obtain E c ∈ F m bias  g − m X i =1 c i A i  ≥    E c ∈ F m bias  g − m X i =1 c i A i     = P (Z( A )) · bias( g | Z( A )) = P (Z( A )) ≥ | F | − dm , where the last inequality applies Theorem 3.2 , using that 0 ∈ Z( A 1 , . . . , A m ) by homogeneity . Th us, there exists a form P = P m i =1 c i A i of degree at most d such that bias( g − P ) ≥ | F | − dm . W e now pro ve Theorem 2 . Pr o of of The or em 2 . Let f : F n → F be a homogeneous p olynomial with k := deg( f ) < 2 d , where F = F q with ch( F q ) > k , and put bias(∆ d f ) = δ . First, w e observ e that bias( f ) ≥ δ 2 d . Indeed, using the monotonicity of Gow ers norms, ∥ g ∥ U k ≥ ∥ g ∥ U d for k ≥ d , and using the fact that, by definition, bias(∆ t f ) = ∥ e ( f ) ∥ 2 t U t for any t , we ha ve that bias( f ) = bias(∆ k f ) = ∥ e ( f ) ∥ 2 k U k ≥ ∥ e ( f ) ∥ 2 k U d = bias(∆ d f ) 2 k − d ≥ δ 2 d . 6 By Theorem 2.1 applied on the k -linear form f , we obtain a partition-rank decomp osition f = r X i =1 Q i R i with Q i , R i m ultilinear forms of degrees at most k − 1, and r ≤ ˜ O k (log p (1 / bias( f ))) = ˜ O d (log q (1 /δ )) . Assume without loss of generalit y that deg( Q i ) ≤ deg( R i ) for ev ery i . Since deg( Q i ) + deg( R i ) = k , w e hav e deg ( Q i ) ≤ k / 2. Th us, f ∈ ⟨ Q 1 , . . . , Q r ⟩ . Plug in the diagonal: let A i ( x ) = Q i ( x, . . . , x ), and observe that f ( x ) = 1 k ! f ( x, . . . , x ), using the homogeneity of f and c h( F ) > k . It follows that f ∈ ⟨ A 1 , . . . , A r ⟩ with each A i homogeneous of degree at most k / 2 < d , b y assumption. W e are now done by Corollary 3.3 : cor 1 and deg( ℓ ) = 1. While f has the decomp osition f = ℓA + ( ℓ + 1) B , this is not a rk ∗ -decomp osition since its degree-1 factors ℓ and ℓ + 1 are not affinely independent. In fact, rk ∗ ( f ) may b e arbitrarily large (see Remark 4.2 next). Remark 4.2. Interestingly , rk ∗ ( f ) can b e more closely related to bias( f ), or rather to analytic rank AR( f ) = − log p (bias( P )), than other standard notions of rank in the literature. F or concreteness, consider the notion used b y Green and T ao in [ GT09 ]: rank( f ) is the least n umber k of p olynomials α 1 , . . . , α k with deg( α i ) < deg( f ) such that f ≡ Γ( α 1 , . . . , α k ) for some function Γ : F k → F . Consider the following p olynomials f : • f ( x, y ) = A ( x ) + ℓ ( y ) with deg ( A ) > deg( ℓ ) = 1. Clearly , rank( f ) ≤ rank( A ) + 1 < ∞ . Ho wev er, rk ∗ ( f ) = ∞ , and moreov er, bias( f ) = bias( A ) bias( ℓ ) = 0, so AR( f ) = ∞ as w ell. 8 Indeed, Z( A ) = { x | A ( x ) = 0 } = { x | S ( x, . . . , x ) = 0 } ⊆ { x | ∆ d f ( x, . . . , x ) = 0 } = Z( f d ). 8 • f ( x, y ) = ( A − B ) ℓ + B with ℓ = ℓ ( y ) of degree 1, and A = A ( x ), B = B ( x ) random of degree d ≥ 2 on n v ariables. Clearly , rank( f ) ≤ 3. In con trast, we claim that both rk ∗ ( f ) and AR( f ) gro w with n . Consider a rk ∗ -decomp osition of f . By affine independence, setting all degree-1 factors to 0 gives a non-empt y affine subspace. Let U b e the non-empty intersection of this affine subspace with the h yp ersurface ℓ = c , for an appropriate c ∈ F . Then co dim( U ) ≤ rk ∗ ( f ) + 1, and the restriction f ′ of f to U has rank( f ′ ) ≤ rk ∗ ( f ). By construction, f ′ is a nonzero linear combination of A and B restricted to U , so rank( f ′ ) gro ws with n , and th us also rk ∗ ( f ). As for AR( f ), w e hav e bias( f ) = E x χ ( B ( x )) E y χ ( ℓ ( y )( A ( x ) − B ( x ))) = E x χ ( B ( x )) 1 A = B ( x ) = P b χ ( b ) P ( B = b, A = b ), so bias( f ) ≈ (1 /q ) P b χ ( b ) P ( B = b ) = (1 /q ) bias( B ), implying that AR( f ) ≈ n , as claimed. 4.1 Prop erties of rk ∗ In general, rk ∗ is not subadditive, unlik e other notions of rank. That is, the sum of tw o rk ∗ - decomp ositions need not be another rk ∗ -decomp osition, for the simple reason that there could b e linear dep endencies b et ween the linear forms in the degree-1 factors of the tw o decompositions. F or example, the rk ∗ of ( x + 1) R and of xS , where deg ( R ) , deg( S ) ≥ 2, is 1, but the rk ∗ of their sum ( x + 1) R + xS need not b e at most 2; in fact, it can b e un b ounded (recall Remark 4.2 ). That b eing said, rk ∗ is subadditive if the tw o p olynomials are of different degrees. Observ ation 4.3 (subadditivity) . rk ∗ ( g + h ) ≤ rk ∗ ( g ) + rk ∗ ( h ) if deg ( h )  = deg( g ) . A basic fact ab out rank is that it is not inherited by partial deriv ativ es: even if a form has lo w rank, a partial deriv ative might not. Consider, sa y , reducible forms; for example, we hav e rk( x · S ) ≤ 1 for any form S = S ( y ), y et the partial deriv ativ e with resp ect to x is S ( y ), which ma y ha ve arbitrarily high rank. That b eing said, we sho w that, p erhaps surprisingly , adding to a polynomial a partial deriv ative do es not increase its rk ∗ . Lemma 4.4 (deriv ation inv ariance) . F or any form g and any dir e ction ve ctor c , rk ∗ ( g + ∂ c g ) ≤ 2 rk( g ) . Com bining the abov e t wo properties, we sho w that rk ∗ satisfies a certain appro ximate “c hain rule”, which w ould complemen t a c hain rule for bias that we prov e later in Section 6 . Prop osition 4.5 (chain rule for rk ∗ ) . F or any p olynomial g + h with top-de gr e e c omp onent g , and any dir e ction ve ctor c , rk ∗ ( g + h ) ≤ 2 rk( g ) + rk ∗ ( h − ∂ c g ) . W e now pro ve the ab o ve prop erties of rk ∗ . Pr o of of Observation 4.3 . Put r 1 = rk ∗ ( g ), r 2 = rk ∗ ( h ). Let g = P r 1 i =1 g i and h = P r 2 i =1 h i , with g i and h i reducible p olynomials with deg( g i ) ≤ deg( g ) and deg( h i ) ≤ deg( h ), b e decomp ositions witnessing rk ∗ ( g ) and rk ∗ ( h ), resp ectively . W e hav e g + h = P r 1 i =1 g i + P r 2 i =1 h i . Assume without loss of generalit y that deg( h ) < deg ( g ). Since deg( h i ) ≤ deg( h ), observ e that the affine indep endence condition for rk ∗ ( g + h ) applies only for the g i , and is therefore satisfied by the definition of rk ∗ ( g ). Th us, rk ∗ ( g + h ) ≤ r 1 + r 2 = rk ∗ ( g ) + rk ∗ ( h ), as needed. 9 Lemma 4.6. F or every g ∈ F orm and a shortest rk -de c omp osition g = P i α i β i , the forms ( α i ) i (and ( β i ) i ) ar e line arly indep endent. Pr o of. Put r = rk( g ), and let g = P r i =1 α i β i b e a rk-decomp osition. Supp ose for contradiction that α r = P r − 1 i =1 c i α i with c i scalars. By homogeneit y , w e may assume that deg( α i ) = deg ( α r ) whenev er c i  = 0. Observe that g has the decomp osition g = P r − 1 i =1 α i ( β i + c i β r ). Each summand is a reducible form, since α i is homogeneous as w ell as β i + c i β r ; indeed, deg ( β i ) = k − deg( α i ) = k − deg( α r ) = deg( β r ) whenever c i  = 0. Th us, r = rk( g ) ≤ r − 1, a contradiction. Next we pro ve the inv ariance of rk ∗ with resp ect to adding a deriv ative. Pr o of of L emma 4.4 . Put r = rk( g ) and k = deg( g ), and write g = P r i =1 α i β i with forms α i , β i satisfying deg ( α i ) ≤ deg( β i ) < k and deg ( α i β i ) = k . By the pro duct rule, ∂ g c = P r i =1 α i ( ∂ c β i ) + ( ∂ c α i ) β i . Observ e that we hav e the following fortunate identit y: g + ∂ c g = r X i =1 ( α i + ∂ c α i )( β i + ∂ c β i ) − r X i =1 ( ∂ c α i )( ∂ c β i ) . No w, since deg ( α i + ∂ c α i ) = deg( α i ) ≤ deg( β i ) = deg( β i + ∂ c β i ), the linear comp onen ts of the factors α i + ∂ c α i are precisely the linear factors α i in the ab o v e decomp osition for g , whic h b y Lemma 4.6 are linearly indep enden t. As for the remaining terms, note that deg (( ∂ c α i )( ∂ c β i )) ≤ deg( α i β i ) − 2 = k − 2 < deg( g + ∂ c g ), so rk ∗ ( g + ∂ c g ) has no requirement on their (linear) factors. W e deduce that the ab o v e (non-homogeneous) decomp osition is a rk ∗ -decomp osition, so rk ∗ ( g + ∂ c g ) ≤ 2 r = 2 rk( g ), as needed. W e are no w ready to pro ve our chain rule for rk ∗ . Pr o of of Pr op osition 4.5 . W e hav e rk ∗ ( g + h ) = rk ∗  ( g + ∂ c g ) + ( h − ∂ c g )  ≤ rk ∗ ( g + ∂ c g ) + rk ∗ ( h − ∂ c g ) ≤ 2 rk( g ) + rk ∗ ( h − ∂ c g ) where the first inequality uses the subadditivity of rk ∗ in Observ ation 4.3 , and the second inequalit y uses Lemma 4.4 . 4.2 Lo w er-degree correlation Recall that standard correlation results sho w that if a p olynomial f has a short decomp osition in terms of low er degree polynomials, then it has a significant correlation with a p olynomial of degree at most deg( f ) − 1. In this section we show that if a p olynomial f has a short rk ∗ -decomp osition, then it has a significan t correlation with a p olynomial of degree at most deg( f ) − 2. This relies on our correlation lemma in Lemma 3.1 , but here we only assume small rk ∗ . Prop osition 4.7 (lo wer-degree correlation) . F or any f ∈ Poly k ( F q ) with k ≥ 3 , cor 2. W e hav e bias( x 2 + x ) · p =    X x ∈ F p ω x 2 + x p    =    ω − 1 / 4 p X x ∈ F p ω ( x +1 / 2) 2 p    =    X y ∈ F p ω y 2 p    = √ p, where the last step uses the form ula for a basic quadratic Gauss sum. Th us, bias( x 2 + x ) = bias( x 2 ) = p − 1 / 2 , whereas for the low er degree comp onen t of x 2 + x we ha ve bias( x ) = 0. Remark 6.2. F or non-homogeneous p olynomials f , having large bias(∆ d f ) do es not guaran tee that the lo wer-degree comp onents of f are structured or biased. F or example, consider f ( x, y ) = h ( x ) · ℓ ( y ) for ℓ ∈ Poly 1 ( F q ) non-homogeneous and h ∈ F orm d ( F q ) with d ≥ 2. The degree- d comp onen t of f is a random p olynomial (in particular, bias( f d ) , bias(∆ d f d ) ≈ q − n ). Ho wev er, a quick calculation gives bias ( f ) ≈ 1 /q , 10 so bias(∆ d f ) ≥ bias( f ) 2 d ≥ q − O d (1) is large. First, we prov e a rather general chain rule for bias. F or an even t E , we henceforth define the c onditional bias as bias ( f | E ) = E ( χ ( f ) | E ). Lemma 6.3 (Chain rule for bias, a general form) . L et A ( x, y ) and B ( y ) b e functions over a finite field, and assume A is line ar in x . F or the event E = { y |∀ x : A ( x, y ) = 0 } , we have bias ( A + B ) = bias ( A ) bias ( B | E ) . 10 bias ( f ) = E x E y χ ( h ( x ) l ( y )) = χ ( c ) E x 1 ( h ( x ) = 0), with c the constant term of ℓ , so bias( f ) = P ( h = 0) ≈ 1 /q . 15 Pr o of. Observ e that for any fixed y ∈ E c , A ( x, y ) + B ( y ) is p olynomial in x of degree exactly 1, and th us takes any v alue in the field with the same probabilit y . F urthermore observe that bias( A ) = E ( χ ( A )) = E y E x ( χ ( A ( x, y ))) = E y 1 E ( y ) = P ( E ) . Therefore, by the law of total exp ectation, bias( A + B ) = E ( χ ( A + B )) = P ( E ) E ( χ ( A + B ) | E ) + P ( E c ) E ( χ ( A + B ) | E c ) = bias( A ) bias( B | E ) + P ( E c ) · 0 . Our main result in this subsection b ounds the bias of the d -times iterated deriv ativ e of f ∈ P oly d +1 in terms of the comp onents of f . Prop osition 6.4. F or f ∈ P oly d +1 ( F ) wher e ch( F )  = 2 , with g = f d +1 and h = f d , bias(∆ d f ) ≤ E c bias( h − ∂ c g ) and bias(∆ d f ) ≤ bias( g ) . Pr o of. By Prop osition 5.1 , ∆ d v g ( x ) = g ( v , x ′ ) where x ′ = x + 1 2 P d t =1 v ( t ) . Thus, ∆ d v f ( x ) = g ( v , x ′ ) + h ( v ), and therefore bias (∆ d v f ( x )) = E x E v χ ( g ( v , x ′ ) + h ( v )) = E x ′ E v χ ( g ( v , x ′ ) + h ( v )) = bias ( g + h ) . This has t wo implications. First, it follows from Lemma 6.3 that bias( g + h ) ≤ bias( g ) . Second, we ha ve bias ( g + h ) = E x E v χ ( g ( v , x ) + h ( v )) = E x E v χ ( ∂ x g ( v ) + h ( v )) = E c E v χ ( ∂ − c g ( v ) + h ( v )) = E c bias ( h − ∂ c g ) . These complete the pro of. 6.2 Main pro of W e now put everything together and pro ve Theorem 1 . Theorem ( 1 ) . F or ev ery f ∈ P oly d +1 ( F q ), with c h( F ) > d + 1, cor d + 1, w e th us ha v e g ( x ) = g ( x d ) / ( d + 1)! and ( h − ∂ c g )( x ) = ( h − ∂ c g )( x d ) /d !, pro ving ( 7 ). By Prop osition 6.4 , bias(∆ d f ) ≤ bias( g ), and bias(∆ d f ) ≤ bias( h − ∂ c g ) for some direction v ector c which w e henceforth fix. Put bias(∆ d f ) = q − r , so that bias( g ) ≥ q − r and bias( h − ∂ c g ) ≥ q − r . (8) No w, com bine ( 7 ), ( 8 ), and Theorem 2.1 to deduce that rk( g ) ≤ rk( g ) ≤ ˜ O d ( r ) and rk( h − ∂ c g ) ≤ rk( h − ∂ c g ) ≤ ˜ O d ( r ) . Apply Prop osition 4.5 to obtain rk ∗ ( g + h ) ≤ 2 rk( g ) + rk( h − ∂ c g ) ≤ ˜ O d ( r ) . Next, apply Prop osition 4.7 with k = d + 1 ( ≥ 3 by assumption) on g + h ∈ P oly d +1 . W e deduce that there exists a p olynomial P with deg( P ) ≤ k − 2 = d − 1 suc h that cor( g + h, P ) ≥ q − 2 rk ∗ ( g + h ) ≥ 1 /q ˜ O d ( r ) . T o finish, write f = g + h + f

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