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https://texercises.com/exercise/polynomials-with-many-variables-over-r/
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Exercise:
fx_...x_r_underlineiin mathcalIc_underlinei x_^i_... x_r^i_rquad c_underlineiin mathbbR underlineii_...i_r textmulti-index mathcalIsubseteq mathbbZ_geq ^r textfinite set mathbbRx_...x_r set of all polynomials in x_...x_r and coeffs in mathbbR. bf Lemma . Suppose that fx_...x_rin mathbbRx_...x_r satisfies that fa_...a_r forall a_...a_rin mathbbR^r. Then fx_...x_r as a polynomial i.e. all the coeffs of fx_...x_r are .

Solution:
Proof. Write fx_...x_r_underlineiin mathcalIc_underlinei x_^i_... x_r^i_r. Let underlineii_...i_rin mathcalI. We'll show that c_underlinei. Since fa_...a_r forall a_...a_rin mathbbR^r then all the partial derivatives of f of any order are all over mathbbR^r Longrightarrow fracpartial^m fpartial x_^i_...partial x_r^i_rBiggr|_x_...x_r mi_+...+i_r. * But fracpartial^m fpartial x_^i_...partial x_r^i_rx_^j_... x_r^j_r cases quad textif exists k:j_k i_k j_j_-...j_-i_+x_^j_-i_... j_rj_r-... j_r-i_r+x_r^j_r-i_r textif forall k i_jgeq i_kcases &Longrightarrow leftfracpartial^kpartial x_^i_...partial x_r^i_rx_^j_... x_r^j_rrightBiggr|_x_...x_r cases quad underlinejneq i i_!... i_r!quad underlinejunderlineicases &Longrightarrow fracpartial^kfpartial x_^i_...partial x_r^i_rBiggr|_x_...x_r textfirst term is because of * c_underlinei &Longrightarrow c_underlinei.
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Exercise:
fx_...x_r_underlineiin mathcalIc_underlinei x_^i_... x_r^i_rquad c_underlineiin mathbbR underlineii_...i_r textmulti-index mathcalIsubseteq mathbbZ_geq ^r textfinite set mathbbRx_...x_r set of all polynomials in x_...x_r and coeffs in mathbbR. bf Lemma . Suppose that fx_...x_rin mathbbRx_...x_r satisfies that fa_...a_r forall a_...a_rin mathbbR^r. Then fx_...x_r as a polynomial i.e. all the coeffs of fx_...x_r are .

Solution:
Proof. Write fx_...x_r_underlineiin mathcalIc_underlinei x_^i_... x_r^i_r. Let underlineii_...i_rin mathcalI. We'll show that c_underlinei. Since fa_...a_r forall a_...a_rin mathbbR^r then all the partial derivatives of f of any order are all over mathbbR^r Longrightarrow fracpartial^m fpartial x_^i_...partial x_r^i_rBiggr|_x_...x_r mi_+...+i_r. * But fracpartial^m fpartial x_^i_...partial x_r^i_rx_^j_... x_r^j_r cases quad textif exists k:j_k i_k j_j_-...j_-i_+x_^j_-i_... j_rj_r-... j_r-i_r+x_r^j_r-i_r textif forall k i_jgeq i_kcases &Longrightarrow leftfracpartial^kpartial x_^i_...partial x_r^i_rx_^j_... x_r^j_rrightBiggr|_x_...x_r cases quad underlinejneq i i_!... i_r!quad underlinejunderlineicases &Longrightarrow fracpartial^kfpartial x_^i_...partial x_r^i_rBiggr|_x_...x_r textfirst term is because of * c_underlinei &Longrightarrow c_underlinei.
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cayley-hamilton, eigenvalue, eigenvector, eth, fs23, lineare algebra, proof
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