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Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)$, where $J_0^m F$ denotes the $m-1$st order Taylor polynomial of $F$ at $0$. Write $$\mathcal P = \bigoplus\limits_{i=0}^{m-1} \mathcal P_i$$ where $\mathcal P_i$ denotes the subspace of homogeneous polynomials of degree $i$. A subspace $V \subset \mathcal P$ is called homogeneous if $$V = \bigoplus\limits_{i=0}^{m-1} V \cap \mathcal P_i.$$

I would like to prove the following. Let $d$ be any reasonable choice of metric on the Grassmannian of $\mathcal P$, for example $d(V, W) = d_H(V \cap B, W \cap B)$ where $d_H$ denotes the Hausdorff distance with respect to the $\ell_2$ norm on the coefficients of $P \in \mathcal P$ and $B$ is the unit ball. For any $\epsilon \in (0, \frac{1}{2})$, there exists $\delta > 0$ such that if $I \subset \mathcal P$ is an ideal and there exists a homogeneous subspace $V \subset \mathcal P$ such that $d(I, V) < \delta$, there exists a homogeneous ideal $J$ such that $d(I, J) < \epsilon$. Furthermore, I would like that $\delta$ is not too small with respect to $\epsilon$, something like $\delta \geq A \epsilon^{D^k}$ where $D = \dim \mathcal P$ and $A, k$ are universal constants.

Without the last caveat, we can prove this by compactness. Suppose there exists $\epsilon > 0$ and ideals $I_n$ and homogeneous subspaces $V_n$ such that $d(I_n, V_n) < \frac{1}{n}$, but for any homogeneous ideal $J$, $d(I, J) \geq \epsilon$. Then, up to a subsequence, $I_n \to I$ and $V_n \to V$, where I is an ideal and $V$ is homogeneous, but then obviously $I = V$ is a homogeneous ideal and this is a contradiction. Is there an explicit construction?

Edit: I want to point out that this can also be approached by applying the Łojasiewicz inequality to the function $f(I) = \text{dist}(I, \mathcal H)$, where $\mathcal H$ is the set of homogeneous subspaces. But it seems that the Łojasiewicz exponent is too large to get the dependence $\delta \sim \exp(\text{poly}(D, \log \epsilon))$ that I want, and even if the exponent were smaller it's not obvious to me how to translate this into an effective bound. I don't think such a broad approach has any hope of giving the estimate I'm looking for.

Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)$, where $J_0^m F$ denotes the $m-1$st order Taylor polynomial of $F$ at $0$. Write $$\mathcal P = \bigoplus\limits_{i=0}^{m-1} \mathcal P_i$$ where $\mathcal P_i$ denotes the subspace of homogeneous polynomials of degree $i$. A subspace $V \subset \mathcal P$ is called homogeneous if $$V = \bigoplus\limits_{i=0}^{m-1} V \cap \mathcal P_i.$$

I would like to prove the following. Let $d$ be any reasonable choice of metric on the Grassmannian of $\mathcal P$, for example $d(V, W) = d_H(V \cap B, W \cap B)$ where $d_H$ denotes the Hausdorff distance with respect to the $\ell_2$ norm on the coefficients of $P \in \mathcal P$ and $B$ is the unit ball. For any $\epsilon \in (0, \frac{1}{2})$, there exists $\delta > 0$ such that if $I \subset \mathcal P$ is an ideal and there exists a homogeneous subspace $V \subset \mathcal P$ such that $d(I, V) < \delta$, there exists a homogeneous ideal $J$ such that $d(I, J) < \epsilon$. Furthermore, I would like that $\delta$ is not too small with respect to $\epsilon$, something like $\delta \geq A \epsilon^{D^k}$ where $D = \dim \mathcal P$ and $A, k$ are universal constants.

Without the last caveat, we can prove this by compactness. Suppose there exists $\epsilon > 0$ and ideals $I_n$ and homogeneous subspaces $V_n$ such that $d(I_n, V_n) < \frac{1}{n}$, but for any homogeneous ideal $J$, $d(I_n, J) \geq \epsilon$. Then, up to a subsequence, $I_n \to I$ and $V_n \to V$, where I is an ideal and $V$ is homogeneous, but then obviously $I = V$ is a homogeneous ideal and this is a contradiction. Is there an explicit construction?

Edit: I want to point out that this can also be approached by applying the Łojasiewicz inequality to the function $f(I) = \text{dist}(I, \mathcal H)$, where $\mathcal H$ is the set of homogeneous subspaces. But it seems that the Łojasiewicz exponent is too large to get the dependence $\delta \sim \exp(\text{poly}(D, \log \epsilon))$ that I want, and even if the exponent were smaller it's not obvious to me how to translate this into an effective bound. I don't think such a broad approach has any hope of giving the estimate I'm looking for.

Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)$, where $J_0^m F$ denotes the $m-1$st order Taylor polynomial of $F$ at $0$. Write $$\mathcal P = \bigoplus\limits_{i=0}^{m-1} \mathcal P_i$$ where $\mathcal P_i$ denotes the subspace of homogeneous polynomials of degree $i$. A subspace $V \subset \mathcal P$ is called homogeneous if $$V = \bigoplus\limits_{i=0}^{m-1} V \cap \mathcal P_i.$$

I would like to prove the following. Let $d$ be any reasonable choice of metric on the Grassmannian of $\mathcal P$, for example $d(V, W) = d_H(V \cap B, W \cap B)$ where $d_H$ denotes the Hausdorff distance with respect to the $\ell_2$ norm on the coefficients of $P \in \mathcal P$ and $B$ is the unit ball. For any $\epsilon \in (0, \frac{1}{2})$, there exists $\delta > 0$ such that if $I \subset \mathcal P$ is an ideal and there exists a homogeneous subspace $V \subset \mathcal P$ such that $d(I, V) < \delta$, there exists a homogeneous ideal $J$ such that $d(I, J) < \epsilon$. Furthermore, I would like that $\delta$ is not too small with respect to $\epsilon$, something like $\delta \geq A \epsilon^{D^k}$ where $D = \dim \mathcal P$ and $A, k$ are universal constants.

Without the last caveat, we can prove this by compactness. Suppose there exists $\epsilon > 0$ and ideals $I_n$ and homogeneous subspaces $V_n$ such that $d(I_n, V_n) < \frac{1}{n}$, but for any homogeneous ideal $J$, $d(I, J) \geq \epsilon$. Then, up to a subsequence, $I_n \to I$ and $V_n \to V$, where I is an ideal and $V$ is homogeneous, but then obviously $I = V$ is a homogeneous ideal and this is a contradiction. Is there an explicit construction?

Edit: I want to point out that this can also be approached by applying the Łojasiewicz inequality to the function $f(I) = \text{dist}(I, \mathcal H)$, where $\mathcal H$ is the set of homogeneous subspaces. But it seems that the Łojasiewicz exponent is too large to get the dependence $\delta \sim \exp(\text{poly}(D))$ that I want, and even if the exponent were smaller it's not obvious to me how to translate this into an effective bound. I don't think such a broad approach has any hope of giving the estimate I'm looking for.

Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)$, where $J_0^m F$ denotes the $m-1$st order Taylor polynomial of $F$ at $0$. Write $$\mathcal P = \bigoplus\limits_{i=0}^{m-1} \mathcal P_i$$ where $\mathcal P_i$ denotes the subspace of homogeneous polynomials of degree $i$. A subspace $V \subset \mathcal P$ is called homogeneous if $$V = \bigoplus\limits_{i=0}^{m-1} V \cap \mathcal P_i.$$

I would like to prove the following. Let $d$ be any reasonable choice of metric on the Grassmannian of $\mathcal P$, for example $d(V, W) = d_H(V \cap B, W \cap B)$ where $d_H$ denotes the Hausdorff distance with respect to the $\ell_2$ norm on the coefficients of $P \in \mathcal P$ and $B$ is the unit ball. For any $\epsilon \in (0, \frac{1}{2})$, there exists $\delta > 0$ such that if $I \subset \mathcal P$ is an ideal and there exists a homogeneous subspace $V \subset \mathcal P$ such that $d(I, V) < \delta$, there exists a homogeneous ideal $J$ such that $d(I, J) < \epsilon$. Furthermore, I would like that $\delta$ is not too small with respect to $\epsilon$, something like $\delta \geq A \epsilon^{D^k}$ where $D = \dim \mathcal P$ and $A, k$ are universal constants.

Without the last caveat, we can prove this by compactness. Suppose there exists $\epsilon > 0$ and ideals $I_n$ and homogeneous subspaces $V_n$ such that $d(I_n, V_n) < \frac{1}{n}$, but for any homogeneous ideal $J$, $d(I, J) \geq \epsilon$. Then, up to a subsequence, $I_n \to I$ and $V_n \to V$, where I is an ideal and $V$ is homogeneous, but then obviously $I = V$ is a homogeneous ideal and this is a contradiction. Is there an explicit construction?

Edit: I want to point out that this can also be approached by applying the Łojasiewicz inequality to the function $f(I) = \text{dist}(I, \mathcal H)$, where $\mathcal H$ is the set of homogeneous subspaces. But it seems that the Łojasiewicz exponent is too large to get the dependence $\delta \sim \exp(\text{poly}(D, \log \epsilon))$ that I want, and even if the exponent were smaller it's not obvious to me how to translate this into an effective bound. I don't think such a broad approach has any hope of giving the estimate I'm looking for.

Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)$, where $J_0^m F$ denotes the $m-1$st order Taylor polynomial of $F$ at $0$. Write $$\mathcal P = \bigoplus\limits_{i=0}^{m-1} \mathcal P_i$$ where $\mathcal P_i$ denotes the subspace of homogeneous polynomials of degree $i$. A subspace $V \subset \mathcal P$ is called homogeneous if $$V = \bigoplus\limits_{i=0}^{m-1} V \cap \mathcal P_i.$$

I would like to prove the following. Let $d$ be any reasonable choice of metric on the Grassmannian of $\mathcal P$, for example $d(V, W) = d_H(V \cap B, W \cap B)$ where $d_H$ denotes the Hausdorff distance with respect to the $\ell_2$ norm on the coefficients of $P \in \mathcal P$ and $B$ is the unit ball. For any $\epsilon \in (0, \frac{1}{2})$, there exists $\delta > 0$ such that if $I \subset \mathcal P$ is an ideal and there exists a homogeneous subspace $V \subset \mathcal P$ such that $d(I, V) < \delta$, there exists a homogeneous ideal $J$ such that $d(I, J) < \epsilon$. Furthermore, I would like that $\delta$ is not too small with respect to $\epsilon$, something like $\delta \geq A \epsilon^{D^k}$ where $D = \dim \mathcal P$ and $A, k$ are universal constants.

Without the last caveat, we can prove this by compactness. Suppose there exists $\epsilon > 0$ and ideals $I_n$ and homogeneous subspaces $V_n$ such that $d(I_n, V_n) < \frac{1}{n}$, but for any homogeneous ideal $J$, $d(I, J) \geq \epsilon$. Then, up to a subsequence, $I_n \to I$ and $V_n \to V$, where I is an ideal and $V$ is homogeneous, but then obviously $I = V$ is a homogeneous ideal and this is a contradiction. Is there an explicit construction?

Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)$, where $J_0^m F$ denotes the $m-1$st order Taylor polynomial of $F$ at $0$. Write $$\mathcal P = \bigoplus\limits_{i=0}^{m-1} \mathcal P_i$$ where $\mathcal P_i$ denotes the subspace of homogeneous polynomials of degree $i$. A subspace $V \subset \mathcal P$ is called homogeneous if $$V = \bigoplus\limits_{i=0}^{m-1} V \cap \mathcal P_i.$$

I would like to prove the following. Let $d$ be any reasonable choice of metric on the Grassmannian of $\mathcal P$, for example $d(V, W) = d_H(V \cap B, W \cap B)$ where $d_H$ denotes the Hausdorff distance with respect to the $\ell_2$ norm on the coefficients of $P \in \mathcal P$ and $B$ is the unit ball. For any $\epsilon \in (0, \frac{1}{2})$, there exists $\delta > 0$ such that if $I \subset \mathcal P$ is an ideal and there exists a homogeneous subspace $V \subset \mathcal P$ such that $d(I, V) < \delta$, there exists a homogeneous ideal $J$ such that $d(I, J) < \epsilon$. Furthermore, I would like that $\delta$ is not too small with respect to $\epsilon$, something like $\delta \geq A \epsilon^{D^k}$ where $D = \dim \mathcal P$ and $A, k$ are universal constants.

Without the last caveat, we can prove this by compactness. Suppose there exists $\epsilon > 0$ and ideals $I_n$ and homogeneous subspaces $V_n$ such that $d(I_n, V_n) < \frac{1}{n}$, but for any homogeneous ideal $J$, $d(I, J) \geq \epsilon$. Then, up to a subsequence, $I_n \to I$ and $V_n \to V$, where I is an ideal and $V$ is homogeneous, but then obviously $I = V$ is a homogeneous ideal and this is a contradiction. Is there an explicit construction?

Edit: I want to point out that this can also be approached by applying the Łojasiewicz inequality to the function $f(I) = \text{dist}(I, \mathcal H)$, where $\mathcal H$ is the set of homogeneous subspaces. But it seems that the Łojasiewicz exponent is too large to get the dependence $\delta \sim \exp(\text{poly}(D))$ that I want, and even if the exponent were smaller it's not obvious to me how to translate this into an effective bound. I don't think such a broad approach has any hope of giving the estimate I'm looking for.